for distillation during training. This requires additional 1.2M or 2.4M
entries for CIFAR10 or other datasets. Our method does not save the previous
model as we do not use distillation.

Note that a large memory consumption usually comes from the memory buffer as
the raw data is of size 32*32*3 or 64*64*3 for CIFAR and T-ImageNet. For
memory buffer of size 2000, a system needs 6.1M or 24.6M entries for CIFAR or
T-ImageNet. Therefore, saving a smaller number of samples is important for
reducing the memory consumption. As we demonstrated in Table 5 and Table 6 in
the main paper, our method performs better than the baselines even with a
smaller memory buffer. In Table 5, we use large memory sizes (e.g., 200 and
2000 for CIFAR10 and other datasets). In Table 6, we reduce the memory size by
half. When we compare the accuracy of our method in Table 6 to those of the
baselines in Table 5, our method still outperforms them on all datasets. Our
method with a smaller memory buffer achieves average classification accuracy
of 88.13, 71.69, 71.29, 64.17, 61.90 on C10-5T, C100-10T, C100-20T, T-5T, and
T-10T. On the other hand, the best baselines achieve 88.98, 69.73, 70.03,
61.03, 58.34 on the same experiments with a larger memory buffer.

## References

  * Abati et al. [2020] Abati, D., Tomczak, J., Blankevoort, T., Calderara, S., Cucchiara, R., Bejnordi, E., 2020\.  Conditional channel gated networks for task-aware continual learning, in: CVPR, pp. 3931–3940.
  * Aljundi et al. [2019] Aljundi, R., Belilovsky, E., Tuytelaars, T., Charlin, L., Caccia, M., Lin, M., Caccia, L., 2019.  Online continual learning with maximal interfered retrieval, in: NeurIPS.
  * Aljundi et al. [2016] Aljundi, R., Chakravarty, P., Tuytelaars, T., 2016.  Expert gate: Lifelong learning with a network of experts.  CoRR, abs/1611.06194 2.
  * Ayub and Wagner [2021] Ayub, A., Wagner, A., 2021.  {EEC}: Learning to encode and regenerate images for continual learning, in: International Conference on Learning Representations.  URL: https://openreview.net/forum?id=lWaz5a9lcFU.
  * Bang et al. [2021] Bang, J., Kim, H., Yoo, Y., Ha, J.W., Choi, J., 2021\.  Rainbow memory: Continual learning with a memory of diverse samples, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8218–8227.
  * Bendale and Boult [2015] Bendale, A., Boult, T., 2015\.  Towards open world recognition, in: Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1893–1902.
  * Buzzega et al. [2020] Buzzega, P., Boschini, M., Porrello, A., Abati, D., CALDERARA, S., 2020.  Dark experience for general continual learning: a strong, simple baseline, in: NeurIPS.
  * Castro et al. [2018] Castro, F.M., Marín-Jiménez, M.J., Guil, N., Schmid, C., Alahari, K., 2018.  End-to-end incremental learning, in: Proceedings of the European conference on computer vision (ECCV).
  * Cha et al. [2021] Cha, H., Lee, J., Shin, J., 2021.  Co2l: Contrastive continual learning, in: ICCV.
  * Chaudhry et al. [2021] Chaudhry, A., Gordo, A., Dokania, P., Torr, P., Lopez-Paz, D., 2021.  Using hindsight to anchor past knowledge in continual learning.  Proceedings of the AAAI Conference on Artificial Intelligence 35, 6993–7001.  URL: https://ojs.aaai.org/index.php/AAAI/article/view/16861.
  * Chaudhry et al. [2018] Chaudhry, A., Ranzato, M., Rohrbach, M., Elhoseiny, M., 2018\.  Efficient lifelong learning with a-gem.  arXiv preprint arXiv:1812.00420 .
  * Chaudhry et al. [2019] Chaudhry, A., Rohrbach, M., Elhoseiny, M., Ajanthan, T., Dokania, P.K., Torr, P.H., Ranzato, M., 2019.  Continual learning with tiny episodic memories .
  * Chen et al. [2020] Chen, T., Kornblith, S., Norouzi, M., Hinton, G., 2020\.  A simple framework for contrastive learning of visual representations, in: ICML.
  * Chen and Liu [2018] Chen, Z., Liu, B., 2018.  Lifelong machine learning.  Morgan & Claypool Publishers.
  * Fei et al. [2016] Fei, G., Wang, S., Liu, B., 2016.  Learning cumulatively to become more knowledgeable, in: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1565–1574.
  * Goodfellow et al. [2015] Goodfellow, I.J., Shlens, J., Szegedy, C., 2015.  Explaining and harnessing adversarial examples.  ICLR .
  * Gummadi et al. [2022] Gummadi, M., Kent, D., Mendez, J.A., Eaton, E., 2022\.  Shels: Exclusive feature sets for novelty detection and continual learning without class boundaries, in: Conference on Lifelong Learning Agents, PMLR. pp. 1065–1085.
  * Han et al. [2019] Han, K., Vedaldi, A., Zisserman, A., 2019.  Learning to discover novel visual categories via deep transfer clustering, in: International Conference on Computer Vision (ICCV).
  * He et al. [2016] He, K., Zhang, X., Ren, S., Sun, J., 2016.  Deep residual learning for image recognition, in: CVPR.
  * Hendrycks and Gimpel [2016] Hendrycks, D., Gimpel, K., 2016\.  A baseline for detecting misclassified and out-of-distribution examples in neural networks.  arXiv preprint arXiv:1610.02136 .
  * Hendrycks et al. [2019] Hendrycks, D., Mazeika, M., Kadavath, S., Song, D., 2019\.  Using self-supervised learning can improve model robustness and uncertainty, in: NeurIPS, pp. 15663–15674.
  * Henning et al. [2021] Henning, C., Cervera, M., D’Angelo, F., Von Oswald, J., Traber, R., Ehret, B., Kobayashi, S., Grewe, B.F., Sacramento, J., 2021.  Posterior meta-replay for continual learning.  Advances in Neural Information Processing Systems 34\.
  * Houlsby et al. [2019] Houlsby, N., Giurgiu, A., Jastrzebski, S., Morrone, B., De Laroussilhe, Q., Gesmundo, A., Attariyan, M., Gelly, S., 2019\.  Parameter-efficient transfer learning for nlp, in: International Conference on Machine Learning, PMLR. pp. 2790–2799.
  * Jafarzadeh et al. [2021] Jafarzadeh, M., Dhamija, A.R., Cruz, S., Li, C., Ahmad, T., Boult, T.E., 2021.  A review of open-world learning and steps toward open-world learning without labels.  arXiv e-prints , arXiv–2011.
  * Karakida and Akaho [2022] Karakida, R., Akaho, S., 2022\.  Learning curves for continual learning in neural networks: Self-knowledge transfer and forgetting, in: International Conference on Learning Representations.
  * Ke et al. [2021] Ke, Z., Liu, B., Ma, N., Xu, H., Shu, L., 2021.  Achieving forgetting prevention and knowledge transfer in continual learning.  Advances in Neural Information Processing Systems 34\.
  * Khosla et al. [2020] Khosla, P., Teterwak, P., Wang, C., Sarna, A., Tian, Y., Isola, P., Maschinot, A., Liu, C., Krishnan, D., 2020.  Supervised contrastive learning.  arXiv preprint arXiv:2004.11362 .
  * Kim et al. [2022] Kim, G., Esmaeilpour, S., Xiao, C., Liu, B., 2022.  Continual learning based on ood detection and task masking, in: CVPR 2022 Workshop on Continual Learning.
  * Kirkpatrick et al. [2017] Kirkpatrick, J., Pascanu, R., Rabinowitz, N., Veness, J., Desjardins, G., Rusu, A.A., Milan, K., Quan, J., Ramalho, T., Grabska-Barwinska, A., Others, 2017.  Overcoming catastrophic forgetting in neural networks.  Proceedings of the National Academy of Sciences 114, 3521–3526.
  * Krizhevsky and Hinton [2009] Krizhevsky, A., Hinton, G., 2009\.  Learning multiple layers of features from tiny images.  Technical Report TR-2009, University of Toronto, Toronto. .
  * Langley [2020] Langley, P., 2020.  Open-world learning for radically autonomous agents, in: Proceedings of the AAAI Conference on Artificial Intelligence, pp. 13539–13543.
  * Le and Yang [2015] Le, Y., Yang, X., 2015.  Tiny imagenet visual recognition challenge.
  * Lee et al. [2018] Lee, K., Lee, K., Lee, H., Shin, J., 2018.  A simple unified framework for detecting out-of-distribution samples and adversarial attacks.  Advances in neural information processing systems 31\.
  * Lee et al. [2019] Lee, K., Lee, K., Shin, J., Lee, H., 2019.  Overcoming catastrophic forgetting with unlabeled data in the wild, in: CVPR.
  * Li and Hoiem [2016] Li, Z., Hoiem, D., 2016.  Learning Without Forgetting, in: ECCV, Springer. pp. 614–629.
  * Liang et al. [2018] Liang, S., Li, Y., Srikant, R., 2018.  Enhancing the reliability of out-of-distribution image detection in neural networks, in: ICLR.  URL: https://openreview.net/forum?id=H1VGkIxRZ.
  * Liu [2020] Liu, B., 2020.  Learning on the job: Online lifelong and continual learning., in: AAAI.
  * Liu et al. [2022] Liu, B., Robertson, E., Grigsby, S., Mazumder, S., 2022\.  Self-initiated open world learning for autonomous ai agents, in: Proceedings of AAAI Symposium on ‘Designing Artificial Intelligence for Open Worlds’.
  * Liu et al. [2020a] Liu, Y., Parisot, S., Slabaugh, G., Jia, X., Leonardis, A., Tuytelaars, T., 2020a.  More classifiers, less forgetting: A generic multi-classifier paradigm for incremental learning, in: ECCV. Springer International Publishing, pp. 699–716.
  * Liu et al. [2020b] Liu, Y., Su, Y., Liu, A.A., Schiele, B., Sun, Q., 2020b.  Mnemonics training: Multi-class incremental learning without forgetting, in: CVPR.
  * Lopez-Paz and Ranzato [2017] Lopez-Paz, D., Ranzato, M., 2017\.  Gradient Episodic Memory for Continual Learning, in: NeurIPS, pp. 6470–6479.
  * Loshchilov and Hutter [2016] Loshchilov, I., Hutter, F., 2016\.  Sgdr: Stochastic gradient descent with warm restarts.  arXiv preprint arXiv:1608.03983 .
  * McCloskey and Cohen [1989] McCloskey, M., Cohen, N.J., 1989\.  Catastrophic interference in connectionist networks: The sequential learning problem, in: Psychology of learning and motivation. Elsevier. volume 24, pp. 109–165.
  * Ostapenko et al. [2019] Ostapenko, O., Puscas, M., Klein, T., Jahnichen, P., Nabi, M., 2019.  Learning to remember: A synaptic plasticity driven framework for continual learning, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 11321–11329.
  * von Oswald et al. [2020] von Oswald, J., Henning, C., Sacramento, J., Grewe, B.F., 2020\.  Continual learning with hypernetworks.  ICLR .
  * Pang et al. [2021] Pang, G., Shen, C., Cao, L., Hengel, A.V.D., 2021.  Deep learning for anomaly detection: A review.  ACM Computing Surveys (CSUR) 54, 1–38.
  * Parmar et al. [2021] Parmar, J., Chouhan, S.S., Raychoudhury, V., Rathore, S.S., 2021\.  Open-world machine learning: Applications, challenges, and opportunities.  arXiv preprint arXiv:2105.13448 .
  * Pentina and Lampert [2014] Pentina, A., Lampert, C., 2014\.  A pac-bayesian bound for lifelong learning, in: International Conference on Machine Learning, PMLR. pp. 991–999.
  * Rajasegaran et al. [2020] Rajasegaran, J., Khan, S., Hayat, M., Khan, F.S., Shah, M., 2020.  itaml: An incremental task-agnostic meta-learning approach, in: CVPR.
  * Ramanujan et al. [2020] Ramanujan, V., Wortsman, M., Kembhavi, A., Farhadi, A., Rastegari, M., 2020.  What’s hidden in a randomly weighted neural network?, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 11893–11902.
  * Rebuffi et al. [2017] Rebuffi, S.A., Kolesnikov, A., Lampert, C.H., 2017.  iCaRL: Incremental classifier and representation learning, in: CVPR, pp. 5533–5542.
  * Russakovsky et al. [2015] Russakovsky, O., Deng, J., Su, H., Krause, J., Satheesh, S., Ma, S., Huang, Z., Karpathy, A., Khosla, A., Bernstein, M., et al., 2015\.  Imagenet large scale visual recognition challenge.  International journal of computer vision 115, 211–252.
  * Scheirer et al. [2014] Scheirer, W.J., Jain, L.P., Boult, T.E., 2014.  Probability models for open set recognition.  IEEE transactions on pattern analysis and machine intelligence 36, 2317–2324.
  * Schölkopf et al. [1999] Schölkopf, B., Williamson, R.C., Smola, A.J., Shawe-Taylor, J., Platt, J.C., et al., 1999.  Support vector method for novelty detection., in: NIPS, Citeseer. pp. 582–588.
  * Serra et al. [2018] Serra, J., Suris, D., Miron, M., Karatzoglou, A., 2018\.  Overcoming catastrophic forgetting with hard attention to the task, in: International Conference on Machine Learning, PMLR. pp. 4548–4557.
  * Shin et al. [2017] Shin, H., Lee, J.K., Kim, J., Kim, J., 2017.  Continual learning with deep generative replay, in: NIPS, pp. 2994–3003.
  * Szegedy et al. [2015] Szegedy, C., Liu, W., Jia, Y., Sermanet, P., Reed, S., Anguelov, D., Erhan, D., Vanhoucke, V., Rabinovich, A., 2015\.  Going deeper with convolutions, in: CVPR.  URL: http://arxiv.org/abs/1409.4842.
  * Tack et al. [2020] Tack, J., Mo, S., Jeong, J., Shin, J., 2020.  Csi: Novelty detection via contrastive learning on distributionally shifted instances, in: NeurIPS.
  * Touvron et al. [2021] Touvron, H., Cord, M., Douze, M., Massa, F., Sablayrolles, A., Jégou, H., 2021\.  Training data-efficient image transformers & distillation through attention, in: International Conference on Machine Learning, PMLR. pp. 10347–10357.
  * van de Ven and Tolias [2019] van de Ven, G.M., Tolias, A.S., 2019\.  Three scenarios for continual learning.  arXiv preprint arXiv:1904.07734 .
  * Wortsman et al. [2020] Wortsman, M., R., V., Liu, R., Kembhavi, A., Rastegari, M., Yosinski, J., Farhadi, A., 2020\.  Supermasks in superposition, in: NeurIPS.
  * Wu et al. [2019] Wu, Y., Chen, Y., Wang, L., Ye, Y., Liu, Z., Guo, Y., Fu, Y., 2019.  Large scale incremental learning, in: CVPR.
  * Xu et al. [2019] Xu, H., Liu, B., Shu, L., Yu, P., 2019.  Open-world learning and application to product classification, in: The World Wide Web Conference, pp. 3413–3419.
  * Yan et al. [2021] Yan, S., Xie, J., He, X., 2021\.  Der: Dynamically expandable representation for class incremental learning, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 3014–3023.
  * Yang et al. [2021] Yang, J., Zhou, K., Li, Y., Liu, Z., 2021.  Generalized out-of-distribution detection: A survey.  arXiv preprint arXiv:2110.11334 .
  * You et al. [2017] You, Y., Gitman, I., Ginsburg, B., 2017.  Large batch training of convolutional networks.  arXiv preprint arXiv:1708.03888 .
  * Zeng et al. [2019] Zeng, G., Chen, Y., Cui, B., Yu, S., 2019.  Continuous learning of context-dependent processing in neural networks.  Nature Machine Intelligence .
  * Zenke et al. [2017] Zenke, F., Poole, B., Ganguli, S., 2017.  Continual learning through synaptic intelligence, in: ICML, pp. 3987–3995.
  * Zhu et al. [2021] Zhu, F., Zhang, X.Y., Wang, C., Yin, F., Liu, C.L., 2021.  Prototype augmentation and self-supervision for incremental learning, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5871–5880.


# The Borel partition spectrum at successors of singular cardinals

Will Brian  W. R. Brian
Department of Mathematics and Statistics
University of North Carolina at Charlotte
9201 University City Blvd.
Charlotte, NC 28223, USA<EMAIL_ADDRESS>wrbrian.wordpress.com

###### Abstract.

Assuming that $0^{\dagger}$ does not exist, we prove that if there is a
partition of $\mathbb{R}$ into $\aleph_{\omega}$ Borel sets, then there is
also a partition of $\mathbb{R}$ into $\aleph_{\omega+1}$ Borel sets.

###### Key words and phrases:

cofinal Kurepa families, singular cardinals, successors of singular cardinals,
Borel sets, partitions

###### 2010 Mathematics Subject Classification:

03E17, 03E35, 54A35

The author is supported by NSF grant DMS-2154229.

##  1\. Introduction

Define the _Borel partition spectrum_ , denoted $\mathfrak{sp}(\text{\small
Borel})$, as follows:

$\mathfrak{sp}(\text{\small
Borel})=\left\\{|\mathcal{P}|\colon\mathcal{P}\text{ is a partition of
}\mathbb{R}\text{ into uncountably many Borel sets}\right\\}\\!.$

Let us begin by reviewing what is known about $\mathfrak{sp}(\text{\small
Borel})$.

  1. (1)

$\aleph_{1}\in\mathfrak{sp}(\text{\small Borel})$.

Hausdorff showed in [13] that $2^{\omega}$, and in fact any uncountable Polish
space, can be expressed as an increasing union
$\bigcup_{\xi<\omega_{1}}E_{\xi}$ of $G_{\delta}$ sets. This implies that
$\mathbb{R}$, or any other uncountable Polish space, can be partitioned into
$\aleph_{1}$ nonempty $F_{\sigma\delta}$ sets.

  2. (2)

$\mathfrak{c}=\max\\!\big{(}\mathfrak{sp}(\text{\small Borel})\big{)}$.

Clearly $\mathfrak{c}\in\mathfrak{sp}(\text{\small Borel})$, because we may
partition $\mathbb{R}$ into singletons. Assuming the Axiom of Choice (which we
do throughout), $\mathbb{R}$ cannot be partitioned into $>\\!|\mathbb{R}|$
sets.

  3. (3)

$\mathfrak{sp}(\text{\small Borel})$ is closed under singular limits.

In other words, if $\lambda$ is singular and $\mathfrak{sp}(\text{\small
Borel})\cap\lambda$ is unbounded in $\lambda$, then
$\lambda\in\mathfrak{sp}(\text{\small Borel})$. This is proved in [4, Theorem
2.8] by adapting a (fairly straightforward) argument of Hechler [14, Theorem
3.4] proving the analogous statement for the so-called MAD spectrum.

In addition to these positive $\mathsf{ZFC}$-provable facts concerning
$\mathfrak{sp}(\text{\small Borel})$, we also have several independence
results, which seem to suggest that very little else can be proved from
$\mathsf{ZFC}$ concerning $\mathfrak{sp}(\text{\small Borel})$.

  1. $(4)$

It is consistent with arbitrary values of $\mathfrak{c}$ that
$\mathfrak{sp}(\text{\small Borel})=\\{\aleph_{1},\mathfrak{c}\\}$.

More precisely, if $\kappa^{\aleph_{0}}=\kappa$ and $\mathsf{GCH}$ holds up to
$\kappa$, then forcing with $\mathrm{Fn}(\kappa,2)$ to add $\kappa$ mutually
generic Cohen reals produces a model in which $\kappa=\mathfrak{c}$ and
$\mathfrak{sp}(\text{\small Borel})=\\{\aleph_{1},\kappa\\}$. This is implicit
in an argument of Miller [23, Section 3]; a more general theorem from which
this follows is proved explicitly by Blass in [1].

  2. $(5)$

It is consistent with arbitrary values of $\mathfrak{c}$ that
$\mathfrak{sp}(\text{\small Borel})=[\aleph_{1},\mathfrak{c}]$.

More precisely, given any $\kappa\geq\mathfrak{c}$ with
$\kappa^{\aleph_{0}}=\kappa$, there is a ccc forcing extension in which
$\mathfrak{c}=\kappa$ and $\mathfrak{sp}(\text{\small
Borel})=[\aleph_{1},\mathfrak{c}]$. This follows from [5, Theorem 3.11], where
the result is proved for $2^{\omega}$ instead of $\mathbb{R}$.

In other words, $(4)$ and $(5)$ say that, regardless of what $\mathfrak{c}$
may be, $\mathfrak{sp}(\text{\small Borel})$ can be as small as possible or as
large as possible (given the requirements in $(1)$ and $(2)$). Furthermore,
the next result shows that $\mathfrak{sp}(\text{\small Borel})$ can be made to
look chaotic and patternless, equal to an almost (but not quite) arbitrary set
of cardinals satisfying $(1)$, $(2)$, and $(3)$.

  1. $(6)$

Suppose $C$ is a set of uncountable cardinals such that:

    1. (a)

$C$ is countable,

    2. (b)

$\aleph_{1}\in C$,

    3. (c)

$C$ has a maximum, and $\max(C)^{\aleph_{0}}=\max(C)$,

    4. (d)

$C$ is closed under singular limits, and

    5. (e)

if $\lambda\in C$ and $\mathrm{cf}(\lambda)=\omega$, then $\lambda^{+}\in C$.

Assuming $\mathsf{GCH}$ holds up to $\max C$, there is a ccc forcing extension
in which $C=\mathfrak{sp}(\text{\small Borel})$.

This is proved as Corollary 3.3 in [4]. An immediate consequence of this
(Corollary 3.4 in [4]) is: for any $A\subseteq\omega\setminus\\{0\\}$ with
$1\in A$, there is a forcing extension in which $\mathfrak{sp}(\text{\small
Borel})=\left\\{\aleph_{n}\colon n\in
A\right\\}\cup\\{\aleph_{\omega},\aleph_{\omega+1}\\}$.

Of the requirements given for $C$ in $(6)$, items (b), (c), and (d) correspond
directly to items $(1)$, $(2)$, and $(3)$ above. These requirements for $C$
cannot be eliminated from $(6)$, because they are necessary features of
$\mathfrak{sp}(\text{\small Borel})$ in any forcing extension. Item (a), on
the other hand, does not represent a necessary feature of
$\mathfrak{sp}(\text{\small Borel})$; we know this because $(5)$ implies that
$\mathfrak{sp}(\text{\small Borel})$ can be uncountable. Item (a) is just an
artifact of the proof of $(6)$ in [4], and it is conceivable that a more
careful or more clever proof could, at some point in the future, eliminate it
from the statement of $(6)$ altogether.

This leaves item (e), which forms the topic of this paper. Our main result is
that, in a sufficiently “$L$-like” set-theoretic universe, (e) represents a
necessary feature of $\mathfrak{sp}(\text{\small Borel})$. Thus, in such a
universe there is nontrivial structure to $\mathfrak{sp}(\text{\small Borel})$
beyond what is stated in items $(1)$, $(2)$, and $(3)$ above.

######  Main Theorem.

Suppose $0^{\dagger}$ does not exist. If $\kappa$ is a singular cardinal with
$\mathrm{cf}(\kappa)=\omega$ and $\kappa\in\mathfrak{sp}(\text{\emph{\small
Borel}})$, then $\kappa^{+}\in\mathfrak{sp}(\text{\emph{\small Borel}})$.

In particular, assuming that $0^{\dagger}$ does not exist, if there is a
partition of $\mathbb{R}$ into $\aleph_{\omega}$ Borel sets then there is also
a partition of $\mathbb{R}$ into $\aleph_{\omega+1}$ Borel sets.

The assertion “$0^{\dagger}$ exists” is a large cardinal axiom a little
stronger (in consistency strength) than the existence of a measurable
cardinal. Roughly, $0^{\dagger}$ bears the same relationship to the $L$-like
models of the form $L[\mu]$, where $\mu$ is a normal measure on a measurable
cardinal, that $0^{\sharp}$ bears to $L$. (See [17] for more.) In particular,
if $0^{\dagger}$ exists then there is an inner model containing a measurable
cardinal. Thus, by contraposition: if
$\aleph_{\omega}\in\mathfrak{sp}(\text{\small Borel})$ while
$\aleph_{\omega+1}\notin\mathfrak{sp}(\text{\small Borel})$, there is an inner
model containing a measurable cardinal.

We present two proofs of the main theorem. Both of them use the assertion
“$0^{\dagger}$ does not exist” in the guise of cofinal Kurepa families. What
these are, and their connection to $0^{\dagger}$, is explained in Section 2.
Then in Sections 3 and 4 we present our two proofs. These proofs use cofinal
Kurepa families in different but clearly analogous ways. The proof in Section
3 is more direct, optimized for both length and clarity. However, the proof in
Section 4 gives slightly more in the end, and it also connects the topic of
this paper to an old Scottish Book problem of Stefan Banach.

##  2\. Preliminaries: cofinal Kurepa families

Given an infinite set $A$, recall that $[A]^{\omega}$ denotes the set of all
countably infinite subsets of $A$.

  * $\circ$

$\mathcal{F}\subseteq[A]^{\omega}$ is _cofinal_ in $[A]^{\omega}$ if for every
$X\in[A]^{\omega}$ there is some $Y\in\mathcal{F}$ such that $Y\supseteq X$.
In other words, $\mathcal{F}\subseteq[A]^{\omega}$ is cofinal if it is cofinal
(in the usual sense of the word) in the poset
$\big{(}[A]^{\mu},\subseteq\big{)}$.

  * $\circ$

$\mathcal{F}\subseteq[A]^{\omega}$ is _Kurepa_ if $\left\\{X\cap Y\colon
Y\in\mathcal{F}\right\\}$ is countable for every countable set $X$.

For example, $\mathcal{F}=[A]^{\omega}$ is cofinal in $[A]^{\omega}$ but not
Kurepa, for any infinite set $A$. If $A$ is uncountable, any countable
$\mathcal{F}\subseteq[A]^{\omega}$ is Kurepa but not cofinal. Generally, if
$\mathcal{F}$ is cofinal then so is every $\mathcal{G}\supseteq\mathcal{F}$,
and if $\mathcal{F}$ is Kurepa then so is every
$\mathcal{G}\subseteq\mathcal{F}$. Thus cofinal families are “large” and
Kurepa families are “small” but in different senses. A subset of
$[A]^{\omega}$ is a _cofinal Kurepa family_ in $[A]^{\omega}$ if it is both
cofinal in $[A]^{\omega}$ and Kurepa.

For example, $\left\\{\alpha\colon\omega\leq\alpha<\omega_{1}\right\\}$ is a
cofinal Kurepa family in $[\omega_{1}]^{\omega}$. (Here, as usual, we adopt
the convention that an ordinal is equal to the set of its predecessors.) Note
that the (non-)existence of cofinal Kurepa families in $[A]^{\omega}$ depends
only on the cardinality of $A$, so in fact this shows there is a cofinal
Kurepa family in $[A]^{\omega}$ for every set $A$ of cardinality $\aleph_{1}$.
It turns out that the same is true for every cardinal below $\aleph_{\omega}$:

######  Theorem 2.1.

If $|A|<\aleph_{\omega}$ then there is a cofinal Kurepa family in
$[A]^{\omega}$.

###### Proof.

See [29, Corollary 7.6.22]. ∎

The situation is more subtle for cardinals $\geq\aleph_{\omega}$, as described
in the following two theorems.

######  Theorem 2.2.

Suppose $\square_{\kappa}$ holds for every uncountable cardinal $\kappa$ with
$\mathrm{cf}(\kappa)=\omega$. Then there is a cofinal Kurepa family in
$[A]^{\omega}$ for every set $A$.

###### Proof.

See [29, Theorem 7.6.26] and the comments following its proof. ∎

######  Theorem 2.3.

It is consistent relative to a huge cardinal that there is no cofinal Kurepa
family in $[\omega_{\omega}]^{\omega}$.

###### Proof.

The generalized Chang Conjecture
$(\aleph_{\omega+1},\aleph_{\omega})\to\to(\aleph_{1},\aleph_{\omega})$
implies there is no cofinal Kurepa family on $\omega_{\omega}$. (This is well
known, and is in fact a relatively straightforward consequence of
$(\aleph_{\omega+1},\aleph_{\omega})\to\to(\aleph_{1},\aleph_{\omega})$. It is
mentioned without proof in [29, Section 7.6], and an indirect proof can be
found in [19, Section 3].) The consistency of
$(\aleph_{\omega+1},\aleph_{\omega})\to\to(\aleph_{1},\aleph_{\omega})$ was
first proved in [20] from a large cardinal hypothesis a little stronger than
the existence of a huge cardinal. Later improvements in [9] reduced the
necessary hypothesis to “just” a huge cardinal. ∎

For every set $A$, define

$\mathrm{cf}[A]^{\mu}\,=\,\min\left\\{|\mathcal{F}|\colon\mathcal{F}\text{ is
cofinal in $[A]^{\omega}$}\right\\}.$

Note that $\mathrm{cf}[A]^{\omega}$ depends only on the cardinality of $A$.

If $\mathcal{F}$ is cofinal in $[A]^{\omega}$ then $\bigcup\mathcal{F}=A$, so
$|A|=\left\lvert\bigcup\mathcal{F}\right\rvert\leq\aleph_{0}\cdot|\mathcal{F}|$.
If $A$ is uncountable, this implies $|\mathcal{F}|\geq|A|$. Thus
$\mathrm{cf}[\kappa]^{\omega}\geq\kappa$ for all uncountable cardinals
$\kappa$. And if $\mathrm{cf}(\kappa)=\omega$, we get an even stronger bound:
a diagonalization argument shows in this case that
$\mathrm{cf}[\kappa]^{\omega}\geq\kappa^{+}$.

_Shelah’s Strong Hypothesis_ , abbreviated $\mathsf{SSH}$, is the following
statement.

  * $\mathsf{SSH}:$

For every uncountable cardinal $\kappa$,

$\mathrm{cf}[\kappa]^{\omega}\,=\,\begin{cases}\kappa&\text{if
}\mathrm{cf}(\kappa)>\omega,\\\ \kappa^{+}&\text{if
}\mathrm{cf}(\kappa)=\omega.\end{cases}$

In other words, $\mathsf{SSH}$ says that $\mathrm{cf}[\kappa]^{\omega}$ should
be as small as possible (subject to the restrictions in the previous
paragraph) for every uncountable cardinal $\kappa$. The original phrasing of
Shelah’s Strong Hypothesis by Shelah is somewhat different: it involves the
pseudo-power function from pcf theory. The statement here labeled
$\mathsf{SSH}$ is equivalent to the original form of Shelah’s Strong
Hypothesis by a theorem of Matet [21].

######  Theorem 2.4.

Suppose $0^{\dagger}$ does not exist. Then $\mathsf{SSH}$ holds, and for every
set $A$ there is a cofinal Kurepa family in $[A]^{\omega}$.

###### Proof.

By work in [15] and [26], if $\square_{\kappa}$ fails for a singular cardinal
$\kappa$, then there is an inner model for a Woodin cardinal. This implies
“$0^{\dagger}$ exists” (with plenty of room to spare). By contraposition, if
$0^{\dagger}$ does not exist then $\square_{\kappa}$ holds for every singular
$\kappa$. By Theorem 2.2, this implies there is a cofinal Kurepa family in
$[A]^{\omega}$ for every set $A$.

It is worth mentioning that the failure of $\square_{\kappa}$ for singular
$\kappa$ is possibly much stronger than the existence of a Woodin cardinal.
(This is already known to be true for $\kappa>\mathfrak{c}$: see [26,
Corollary 5]. But in the context of this paper, we are more concerned with
$\square_{\kappa}$ for $\kappa<\mathfrak{c}$.) To quote [26], “the extent to
which we can $\dots$ obtain lower bounds on the large cardinal consistency
strength of the failure of $\square_{\kappa}$ $\dots$ is limited by our
ability to construct core models and prove covering theorems for them.”

The bottleneck for this theorem is not the existence of cofinal Kurepa
families, which follows from $\square_{\kappa}$ holding for singular $\kappa$,
but rather $\mathsf{SSH}$. The exact consistency strength of
$\neg\mathsf{SSH}$ is not known. The failure of $\mathsf{SSH}$ at some
$\kappa>\mathfrak{c}$ is equivalent to the failure of the Singular Cardinals
Hypothesis, $\mathsf{SCH}$. The exact consistency strength of this is known,
due to work of Gitik [11, 12]: it is equiconsistent with the existence of a
measurable cardinal $\mu$ with Mitchell order $\mu^{++}$.

Gitik’s work makes use of the so-called weak covering lemma holding over inner
models of $\mathsf{GCH}$. The weak covering lemma does not seem to tell us
anything about $\mathsf{SSH}$ for cardinals $\kappa<\mathfrak{c}$. However,
$\mathsf{SSH}$ follows from the full covering lemma holding over an inner
model of $\mathsf{GCH}$. (This was first observed in [16, Lemmas 4.9 and
4.10].)

By work of Dodd and Jensen [6], if there is no inner model containing a
measurable cardinal, then the covering lemma holds over an inner model of
$\mathsf{GCH}$ (specifically, the Dodd-Jensen core model $K^{\mathrm{DJ}}$).
By further work of Dodd and Jensen [7], if there is an inner model containing
a measurable cardinal but $0^{\dagger}$ does not exist, then the covering
lemma holds over an inner model of $\mathsf{GCH}$ (specifically, either
$L[\mu]$ for some measure $\mu$ with $\mathrm{crit}(\mu)$ as small as
possible, or $L[\mu,C]$ for some sequence $C$ Prikry-generic over $L[\mu]$).
For more information, see Mitchell’s article [24].

Either way, if $0^{\dagger}$ does not exist then Jensen’s covering lemma holds
over an inner model M of $\mathsf{GCH}$, and $\mathsf{SCH}$ follows by the
results in [16]. ∎

To close this section, we include one more fact concerning cofinal Kurepa
families that will be used in what follows.

######  Lemma 2.5.

Every cofinal Kurepa family in $[A]^{\omega}$ has cardinality
$\mathrm{cf}[A]^{\omega}$.

###### Proof.

Suppose $\mathcal{F}$ is a cofinal Kurepa family in $[A]^{\omega}$. Because
$\mathcal{F}$ is cofinal, $|\mathcal{F}|\geq\mathrm{cf}[A]^{\omega}$. To prove
the reverse inequality, fix some cofinal $\mathcal{C}\subseteq[A]^{\omega}$
with $|\mathcal{C}|=\mathrm{cf}[A]^{\omega}$.

For every $X\in\mathcal{F}$, choose a set $c(X)\in\mathcal{C}$ with
$c(X)\supseteq X$ (which can be done because $\mathcal{C}$ is cofinal). If
$|\mathcal{C}|=\mathrm{cf}[A]^{\omega}<|\mathcal{F}|$, then by the pigeonhole
principle there is some particular $Y\in\mathcal{C}$ such that
$\left\\{X\in\mathcal{F}\colon c(X)=Y\right\\}$ is uncountable. But then
$\left\\{X\cap Y\colon
X\in\mathcal{F}\right\\}\supseteq\left\\{X\in\mathcal{F}\colon
c(X)=Y\right\\}$ is uncountable, contradicting our assumption that
$\mathcal{F}$ is Kurepa. ∎

######  Corollary 2.6.

Assuming $0^{\dagger}$ does not exist, if $\kappa$ is an uncountable cardinal
with $\mathrm{cf}(\kappa)=\omega$ and $|A|=\kappa$, then there is a cofinal
Kurepa family in $[A]^{\omega}$ with cardinality $\kappa^{+}$.

###### Proof.

This follows immediately from Theorem 2.4 and Lemma 2.5. ∎

##  3\. A proof of the main theorem

We begin this section by observing that the definition of
$\mathfrak{sp}(\text{\small Borel})$ does not depend on $\mathbb{R}$: it
remains unchanged when $\mathbb{R}$ is replaced by any other uncountable
Polish space. This is proved already as Theorem 2.1 in [4], but the proof is
short, and is used in the proof of our main theorem, so we reproduce the
argument here.

######  Lemma 3.1.

If $X$ is any uncountable Polish space, then

$\mathfrak{sp}(\text{\emph{\small
Borel}})=\left\\{\kappa>\aleph_{0}\colon\text{there is a partition of }X\text{
into }\kappa\text{ Borel sets}\right\\}.$

###### Proof.

By a theorem of Kuratowski (see [18, Theorem 15.6]), any two uncountable
Polish spaces are Borel isomorphic: in other words, there is a bijection
$f:\mathbb{R}\to X$ such that $A\subseteq\mathbb{R}$ is Borel if and only if
$f[A]$ is Borel. Thus if $\mathcal{P}$ is any partition of $\mathbb{R}$ into
Borel sets, then $\left\\{f[B]\colon B\in\mathcal{P}\right\\}$ is a partition
of $X$ into Borel sets, and if $\mathcal{Q}$ is any partition of $X$ into
Borel sets, then $\left\\{f^{-1}[B]\colon B\in\mathcal{Q}\right\\}$ is a
partition of $\mathbb{R}$ into Borel sets. ∎

######  Theorem 3.2.

Let $\kappa$ be a cardinal and suppose there is a cofinal Kurepa family in
$[\kappa]^{\omega}$. If $\kappa\in\mathfrak{sp}(\text{\emph{\small Borel}})$,
then $\mathrm{cf}[\kappa]^{\omega}\in\mathfrak{sp}(\text{\emph{\small
Borel}})$.

###### Proof.

Suppose $\kappa\in\mathfrak{sp}(\text{\small Borel})$, and let $\mathcal{P}$
be a partition of $\mathbb{R}$ into $\kappa$ Borel sets. Because there is a
cofinal Kurepa family in $[\kappa]^{\omega}$ and $|\mathcal{P}|=\kappa$, there
is a cofinal Kurepa family in $[\mathcal{P}]^{\omega}$. Let $\mathcal{K}$ be
some such family.

To prove the theorem, it suffices (by Lemmas 3.1 and 2.5) to show that there
is a partition of $\mathbb{R}^{\omega}$ into
$\left\lvert\mathcal{K}\right\rvert$ Borel sets. We adopt the convention that
points in $\mathbb{R}^{\omega}$ are functions $\omega\to\mathbb{R}$, so that
$x(n)$ denotes the $n^{\mathrm{th}}$ coordinate of $x$ for any given
$x\in\mathbb{R}^{\omega}$.

For every $\mathcal{A}\in\mathcal{K}$, define

$X_{\mathcal{A}}=(\textstyle\bigcup\mathcal{A})^{\omega}=\left\\{x\in\mathbb{R}^{\omega}\colon
x(n)\in\textstyle\bigcup\mathcal{A}\text{ for all }n\in\omega\right\\}.$

Note that, because each $\mathcal{A}\in\mathcal{K}$ is a countable collection
of Borel subsets of $\mathbb{R}$, each $X_{\mathcal{A}}$ is a Borel subset of
$\mathbb{R}^{\omega}$.

We claim that
$\bigcup\left\\{X_{\mathcal{A}}\colon\mathcal{A}\in\mathcal{K}\right\\}=\mathbb{R}^{\omega}$.
To see this, let $x\in\mathbb{R}^{\omega}$. For each $n\in\omega$, there is
some $B_{n}\in\mathcal{P}$ with $x\in B_{n}$. Because $\mathcal{K}$ is cofinal
in $[\mathcal{P}]^{\omega}$, there is some $\mathcal{A}\in\mathcal{K}$ with
$\left\\{B_{n}\colon n\in\omega\right\\}\subseteq\mathcal{A}$. This implies
$x\in X_{\mathcal{A}}$.

Let $\prec$ be a well ordering of $\mathcal{K}$, and for each
$\mathcal{A}\in\mathcal{K}$ define

$Y_{\mathcal{A}}=X_{\mathcal{A}}\setminus\textstyle\bigcup\left\\{X_{\mathcal{B}}\colon\mathcal{B}\prec\mathcal{A}\right\\}.$

Because
$\bigcup\left\\{X_{\mathcal{A}}\colon\mathcal{A}\in\mathcal{K}\right\\}=\mathbb{R}^{\omega}$,
the $Y_{\mathcal{A}}$’s form a partition of $\mathbb{R}^{\omega}$, or more
precisely (as some of the $Y_{\mathcal{A}}$’s may be empty),
$\mathcal{Q}=\left\\{Y_{\mathcal{A}}\colon\mathcal{A}\in\mathcal{K}\right\\}\setminus\\{\emptyset\\}$
is a partition of $\mathbb{R}^{\omega}$. To prove the theorem, it remains to
show that $\left\lvert\mathcal{Q}\right\rvert=|\mathcal{K}|$, and that every
$Y_{\mathcal{A}}$ is a Borel subset of $\mathbb{R}^{\omega}$.

To see that $\left\lvert\mathcal{Q}\right\rvert=|\mathcal{K}|$, it suffices to
show that $Y_{\mathcal{A}}\neq\emptyset$ for $|\mathcal{K}|$-many
$\mathcal{A}$. (This is because it is clear from the definition that
$Y_{\mathcal{A}}\cap Y_{\mathcal{B}}=\emptyset$ whenever
$\mathcal{A}\neq\mathcal{B}$; the only non-injectivity in the map
$\mathcal{A}\mapsto Y_{\mathcal{A}}$ is that it may map many sets to
$\emptyset$.) For every $\mathcal{A}\in\mathcal{K}$, define

$\mathcal{Z}_{\mathcal{A}}=\left\\{B\in\mathcal{P}\colon x(n)\in B\text{ for
some }x\in Y_{\mathcal{A}}\text{ and some }n\in\omega\right\\}.$

Note that $\mathcal{Z}_{\mathcal{A}}=\emptyset$ whenever
$Y_{\mathcal{A}}=\emptyset$. For every $\mathcal{A}\in\mathcal{K}$, we have
$Y_{\mathcal{A}}\subseteq X_{\mathcal{A}}$, which means that

$\mathcal{Z}_{\mathcal{A}}\subseteq\left\\{B\in\mathcal{P}\colon x(n)\in
B\text{ for some }x\in X_{\mathcal{A}}\text{ and some
}n\in\omega\right\\}=\mathcal{A}.$

In particular,
$\left\\{\mathcal{Z}_{\mathcal{A}}\colon\mathcal{A}\in\mathcal{K}\right\\}$ is
a collection of countable subsets of $\mathcal{P}$. We claim this collection
is cofinal in $[\mathcal{P}]^{\omega}$. To see this, let $\left\\{B_{n}\colon
n\in\omega\right\\}$ be an arbitrary countable subset of $\mathcal{P}$, and
for each $n\in\omega$ fix some $x_{n}\in B_{n}$. Consider the point
$x\in\mathbb{R}^{\omega}$ with $x(n)=x_{n}$ for all $n\in\omega$. Because
$\mathcal{Q}$ is a partition of $\mathbb{R}^{\omega}$, there is some
$\mathcal{A}\in\mathcal{K}$ with $x\in Y_{\mathcal{A}}$. It follows that
$\left\\{B_{n}\colon n\in\omega\right\\}\subseteq\mathcal{Z}_{\mathcal{A}}$.
As $\left\\{B_{n}\colon n\in\omega\right\\}$ was an arbitrary member of
$[\mathcal{P}]^{\omega}$, this shows
$\left\\{\mathcal{Z}_{\mathcal{A}}\colon\mathcal{A}\in\mathcal{K}\right\\}$ is
cofinal in $[\mathcal{P}]^{\omega}$. Hence
$\left\lvert\left\\{\mathcal{Z}_{\mathcal{A}}\colon\mathcal{A}\in\mathcal{K}\right\\}\right\rvert\geq\mathrm{cf}[\mathcal{P}]^{\omega}=|\mathcal{K}|$,
which means
$\left\lvert\left\\{\mathcal{Z}_{\mathcal{A}}\colon\mathcal{A}\in\mathcal{K}\right\\}\right\rvert=|\mathcal{K}|$.
In particular, $\mathcal{Z}_{\mathcal{A}}\neq\emptyset$ for
$|\mathcal{K}|$-many $\mathcal{A}\in\mathcal{K}$, which implies
$Y_{\mathcal{A}}\neq\emptyset$ for $|\mathcal{K}|$-many
$\mathcal{A}\in\mathcal{K}$. Hence $|\mathcal{Q}|=|\mathcal{K}|$.

To see that each $Y_{\mathcal{A}}$ is Borel in $\mathbb{R}^{\omega}$, we use
the Kurepa property of $\mathcal{K}$. Fix $\mathcal{A}\in\mathcal{K}$. Because
$\mathcal{K}$ is Kurepa, there is a countable
$\mathcal{F}\subseteq\mathcal{K}$ such that
$\left\\{\mathcal{A}\cap\mathcal{B}\colon\mathcal{B}\in\mathcal{K}\right\\}=\left\\{\mathcal{A}\cap\mathcal{B}\colon\mathcal{B}\in\mathcal{F}\right\\}$.
This implies

$\left\\{\mathcal{A}\cap\mathcal{B}\colon\mathcal{B}\in\mathcal{K}\text{ and
}\mathcal{B}\prec\mathcal{A}\right\\}=\left\\{\mathcal{A}\cap\mathcal{B}\colon\mathcal{B}\in\mathcal{G}\right\\}$

for some (countable) $\mathcal{G}\subseteq\mathcal{F}$. In particular,

$\displaystyle Y_{\mathcal{A}}$ $\displaystyle=\textstyle
X_{\mathcal{A}}\setminus\bigcup\left\\{X_{\mathcal{B}}\colon\mathcal{B}\prec\mathcal{A}\right\\}$
$\displaystyle=\textstyle(\bigcup\mathcal{A})^{\omega}\setminus\bigcup\left\\{(\bigcup\mathcal{B})^{\omega}\colon\mathcal{B}\prec\mathcal{A}\right\\}$
$\displaystyle=\textstyle(\bigcup\mathcal{A})^{\omega}\setminus\bigcup\left\\{(\bigcup(\mathcal{B}\cap\mathcal{A}))^{\omega}\colon\mathcal{B}\prec\mathcal{A}\right\\}$
$\displaystyle=\textstyle(\bigcup\mathcal{A})^{\omega}\setminus\bigcup\left\\{(\bigcup(\mathcal{B}\cap\mathcal{A}))^{\omega}\colon\mathcal{B}\in\mathcal{G}\right\\}$
$\displaystyle=\textstyle
X_{\mathcal{A}}\setminus\bigcup\left\\{X_{\mathcal{B}}\colon\mathcal{B}\in\mathcal{G}\right\\}.$

Because $X_{\mathcal{A}}$ and all of the $X_{\mathcal{B}}$’s are Borel sets in
$\mathbb{R}^{\omega}$, and because $\mathcal{G}$ is countable, it follows that
$Y_{\mathcal{A}}$ is Borel. ∎

######  Theorem 3.3.

Assume $0^{\dagger}$ does not exist. If $\kappa$ is a singular cardinal with
$\mathrm{cf}(\kappa)=\omega$ and $\kappa\in\mathfrak{sp}(\text{\emph{\small
Borel}})$, then $\kappa^{+}\in\mathfrak{sp}(\text{\emph{\small Borel}})$.

###### Proof.

This follows immediately from Corollary 2.6 and Theorem 3.2. ∎

This completes our first proof of the main theorem. Note that the hypothesis
“$0^{\dagger}$ does not exist” can be replaced with the possibly weaker
hypothesis “$\mathrm{cf}[\kappa]^{\omega}=\kappa^{+}$ and there is a cofinal
Kurepa family in $[\kappa]^{\omega}$.”

######  Question 3.4.

Is it consistent, relative to some large cardinal hypothesis, that there is a
singular cardinal $\lambda$ with $\mathrm{cf}(\lambda)=\omega$ such that
$\lambda\in\mathfrak{sp}(\text{\emph{\small Borel}})$ but
$\lambda^{+}\notin\mathfrak{sp}(\text{\emph{\small Borel}})$?

######  Question 3.5.

In particular, is it consistent relative to some large cardinal hypothesis
that $\aleph_{\omega}\in\mathfrak{sp}(\text{\emph{\small Borel}})$ but
$\aleph_{\omega+1}\notin\mathfrak{sp}(\text{\emph{\small Borel}})$?

######  Question 3.6.

Is it consistent, relative to some large cardinal hypothesis, that there is a
singular cardinal $\lambda$ with $\mathrm{cf}(\lambda)=\omega$ such that
$\lambda\in\mathfrak{sp}(\text{\emph{\small Borel}})$ but
$\mathrm{cf}[\lambda]^{\omega}\notin\mathfrak{sp}(\text{\emph{\small
Borel}})$?

##  4\. A second proof of the main theorem

In this section we give our second proof of the main theorem. This second
proof has a more topological flavor than the one presented in the previous
section. Throughout this section, all ordinals are considered to have the
discrete topology. Thus, for example, $\kappa^{\omega}$ is a completely
metrizable space, sometimes called the Baire space of weight $\kappa$ (e.g.,
in [8, Example 4.2.12]).

Given a topological space $X$, define

$\mathfrak{par}(X)=\min\left\\{|\mathcal{P}|\colon\mathcal{P}\text{ is a
partition of }X\text{ into Polish spaces}\right\\}.$

Note that $\mathfrak{par}(X)$ is well-defined and $\leq\\!|X|$ for every space
$X$, because we may partition $X$ into singletons. The invariant
$\mathfrak{par}(X)$ was introduced and studied in [3], the main idea being to
compare $\mathfrak{par}(X)$ with $\mathfrak{cov}(X)$ (defined as the least
size of a covering of $X$ with Polish spaces) when $X$ is completely
metrizable.

The next two lemmas, which are implicit in [2] and [5], allow us to transfer
facts about the spaces $\kappa^{\omega}$ and $\mathfrak{par}(\kappa^{\omega})$
to facts about $\mathfrak{sp}(\text{\small Borel})$.

######  Lemma 4.1.

If there is a continuous bijection $X\to\mathbb{R}$, for some space $X$, then
there is a partition of $\mathbb{R}$ into $\mathfrak{par}(X)$ Borel sets.

###### Proof.

Suppose $f:X\to\mathbb{R}$ is a continuous bijection, and let $\mathcal{P}$ be
a partition of $X$ into $\mathfrak{par}(X)$ Polish spaces. By a theorem of
Lusin and Suslin [18, Theorem 15.1], if $A$ is Polish and $g:A\to\mathbb{R}$
is a continuous injection, then $g[A]$ is Borel in $\mathbb{R}$ (in fact,
$g[B]$ is Borel in $\mathbb{R}$ for every Borel $B\subseteq A$). In
particular, $f\\!\restriction\\!A$ is a continuous injection $A\to\mathbb{R}$
for every $A\in\mathcal{P}$, which means $f[A]$ is Borel in $\mathbb{R}$ for
every $A\in\mathcal{P}$. Hence $\left\\{f[A]\colon A\in\mathcal{P}\right\\}$
is a partition of $\mathbb{R}$ into Borel sets. ∎

######  Lemma 4.2.

Let $\kappa$ be an infinite cardinal. If there is a partition of $\mathbb{R}$
into $\kappa$ Borel sets, then there is a continuous bijection
$\kappa^{\omega}\to\mathbb{R}$.

###### Proof.

Suppose $\kappa\in\mathfrak{sp}(\text{\small Borel})$. By Lemma 3.1, this
implies there is a partition $\mathcal{P}$ of the Baire space
$\omega^{\omega}$ into $\kappa$ Borel sets.

By a theorem of Lusin and Suslin [18, Theorem 13.7], every Borel subset of a
Polish space is the bijective continuous image of a closed subset of
$\omega^{\omega}$. In particular, for every $A\in\mathcal{P}$, there is a
closed $F_{A}\subseteq\omega^{\omega}$ and a continuous bijection
$f_{A}:F_{A}\to A$. This implies that the map $(x,y)\mapsto(f_{A}(x),y)$ is a
continuous bijection $F_{A}\times\omega^{\omega}\to A\times\omega^{\omega}$.
By the Alexandroff-Urysohn characterization of the Baire space
$\omega^{\omega}$ [8, Exercise 7.2.G],
$F_{A}\times\omega^{\omega}\approx\omega^{\omega}$. Thus for every
$A\in\mathcal{P}$, there is a continuous bijection $g_{A}:\omega^{\omega}\to
A\times\omega^{\omega}$.

Let $\mathcal{Q}=\left\\{A\times\omega^{\omega}\colon
A\in\mathcal{P}\right\\}$; this is a partition of
$\omega^{\omega}\times\omega^{\omega}$ into $\kappa$ Borel sets. Taking the
disjoint union of the mappings $g_{A}$ from the previous paragraph, we obtain
a continuous bijection
$G:\kappa\times\omega^{\omega}\to\omega^{\omega}\times\omega^{\omega}$.

Let $G^{\omega}$ denote the mapping
$(x_{0},x_{1},x_{2},\dots)\mapsto(G(x_{0}),G(x_{1}),G(x_{2}),\dots)$
(sometimes called the “diagonal mapping”). Because $G$ is a continuous
bijection
$\kappa\times\omega^{\omega}\to\omega^{\omega}\times\omega^{\omega}$, the
diagonal mapping $G^{\omega}$ is a continuous bijection
$(\kappa\times\omega^{\omega})^{\omega}\to(\omega^{\omega}\times\omega^{\omega})^{\omega}$.
But $(\kappa\times\omega^{\omega})^{\omega}\approx\kappa^{\omega}$ and
$(\omega^{\omega}\times\omega^{\omega})^{\omega}\approx\omega^{\omega}$ (both
of these facts follow from [8, Exercise 7.2.G]), so this shows there is a
continuous bijection $\kappa^{\omega}\to\omega^{\omega}$.

To finish the proof, simply compose this continuous bijection
$\kappa^{\omega}\to\omega^{\omega}$ with a continuous bijection
$\omega^{\omega}\to\mathbb{R}$. (Recall that every Polish space without
isolated points is a continuous bijective image of $\omega^{\omega}$ [18,
Exercise 7.15].) ∎

######  Theorem 4.3.

Let $\kappa$ be an uncountable cardinal. If
$\kappa\in\mathfrak{sp}(\text{\small Borel})$, then
$\mathfrak{par}(\kappa^{\omega})\in\mathfrak{sp}(\text{\small Borel})$ also.

###### Proof.

This follows immediately from Lemmas 4.1 and 4.2. ∎

Note the similarity between Theorem 3.2 and Theorem 4.3. Our next theorem
shows that $\mathfrak{par}(\kappa^{\omega})=\mathrm{cf}[\kappa]^{\omega}$ if
there is a cofinal Kurepa family on $\kappa$, which will complete our second
proof of Theorem 3.2 and the main theorem (which follows directly from it). We
note that the equality
$\mathfrak{par}(\kappa^{\omega})=\mathrm{cf}[\kappa]^{\omega}$ need not be
true in general: it is consistent relative to a huge cardinal to have
$\mathfrak{par}(\kappa^{\omega})>\mathrm{cf}[\kappa]^{\omega}$ (see [3,
Section 3]).

######  Theorem 4.4.

Let $\kappa$ be an uncountable cardinal. If there is a cofinal Kurepa family
on $\kappa$, then
$\mathfrak{par}(\kappa^{\omega})=\mathrm{cf}[\kappa]^{\omega}$.

The following proof contains some ideas similar to what is found in [3,
Sections 2 and 4], where what is called Corollary4.5 below was first proved.
Something similar to the proof of Theorem 4.4 is implicit in [3], though
Jensen matrices are used there rather than Kurepa families. However, the
arguments in [3] are aimed at proving something rather different, and they
feature several ideas that are not relevant here. What follows is a relatively
short proof, modifying and distilling from [3] what is needed for our analysis
of $\mathfrak{sp}(\text{\small Borel})$.

###### Proof of Theorem 4.4.

Let $\mathcal{K}$ be a cofinal Kurepa family on $\kappa$. To prove the
theorem, it suffices, by Lemma 2.5, to show there is a partition of
$\kappa^{\omega}$ into $\left\lvert\mathcal{K}\right\rvert$ Polish spaces.
Recall that a subspace of a completely metrizable space is itself completely
metrizable if and only if it is $G_{\delta}$. Thus a subspace of
$\kappa^{\omega}$ is Polish if and only if it is both second countable and
$G_{\delta}$.

For every $A\in\mathcal{K}$, note that $A^{\omega}$ is second countable
(because $A$ is countable) and closed in $\kappa^{\omega}$. Furthermore, we
claim $\kappa^{\omega}$ is covered by the sets of this form: i.e.,
$\bigcup\left\\{A^{\omega}\colon A\in\mathcal{K}\right\\}=\kappa^{\omega}$. To
see this, let $x\in\kappa^{\omega}$. As $\mathcal{K}$ is cofinal in
$[\kappa]^{\omega}$, there is some $A\in\mathcal{K}$ with $\left\\{x(n)\colon
n\in\omega\right\\}\subseteq A$, and this means $x\in A^{\omega}$. As $x$ was
arbitrary, $\bigcup\left\\{A^{\omega}\colon
A\in\mathcal{K}\right\\}=\kappa^{\omega}$.

Let $\prec$ be a well ordering of $\mathcal{K}$, and for each
$A\in\mathcal{K}$ define

$Y_{A}=A^{\omega}\setminus\textstyle\bigcup\left\\{B^{\omega}\colon B\prec
A\right\\}.$

Because $\bigcup\left\\{A^{\omega}\colon
A\in\mathcal{K}\right\\}=\kappa^{\omega}$, the $Y_{A}$’s form a partition of
$\kappa^{\omega}$, or more precisely (because some of the $Y_{A}$ may be
empty), $\mathcal{Q}=\left\\{Y_{A}\colon
A\in\mathcal{K}\right\\}\setminus\\{\emptyset\\}$ is a partition of
$\kappa^{\omega}$. To prove the theorem, it remains to show that
$\left\lvert\mathcal{Q}\right\rvert=|\mathcal{K}|$, and that every
$Y_{A}\in\mathcal{Q}$ is Polish.

To see that $\left\lvert\mathcal{Q}\right\rvert=\mathrm{cf}[\kappa]^{\omega}$,
define for every $A\in\mathcal{K}$

$\mathcal{Z}_{A}=\left\\{\alpha\in\kappa\colon x(n)=\alpha\text{ for some
}x\in Y_{A}\text{ and some }n\in\omega\right\\}.$

Note that we have $\mathcal{Z}_{A}=\emptyset$ whenever $Y_{A}=\emptyset$. For
every $A\in\mathcal{K}$, we have $Y_{A}\subseteq X_{A}$, which means that

$\mathcal{Z}_{A}\subseteq\left\\{\alpha\in\kappa\colon x(n)=\alpha\text{ for
some }x\in A^{\omega}\text{ and some }n\in\omega\right\\}=A.$

In particular, $\left\\{\mathcal{Z}_{A}\colon A\in\mathcal{K}\right\\}$ is a
collection of countable subsets of $\kappa$. We claim this collection is
cofinal in $[\kappa]^{\omega}$. To see this, let $\left\\{x_{n}\colon
n\in\omega\right\\}$ be an arbitrary countable subset of $\kappa$, and define
$x\in\kappa^{\omega}$ by setting $x(n)=x_{n}$ for all $n\in\omega$. Because
$\mathcal{Q}$ is a partition of $\kappa^{\omega}$, there is some
$A\in\mathcal{K}$ with $x\in Y_{A}$. It follows that $\left\\{x_{n}\colon
n\in\omega\right\\}\subseteq\mathcal{Z}_{A}$. As $\left\\{x_{n}\colon
n\in\omega\right\\}$ was arbitrary, it follows that
$\left\\{\mathcal{Z}_{A}\colon A\in\mathcal{K}\right\\}$ is cofinal in
$[\kappa]^{\omega}$. Hence $\left\lvert\left\\{\mathcal{Z}_{A}\colon
A\in\mathcal{K}\right\\}\right\rvert\geq\mathrm{cf}[\kappa]^{\omega}=|\mathcal{K}|$.
In particular, $\mathcal{Z}_{A}\neq\emptyset$ for $|\mathcal{K}|$-many
$A\in\mathcal{K}$. Consequently, $Y_{A}\neq\emptyset$ for $|\mathcal{K}|$-many
$A\in\mathcal{K}$. As the $Y_{A}$ are disjoint, this implies
$\left\lvert\mathcal{Q}\right\rvert=|\mathcal{K}|$.

To see that each $Y_{A}$ is Polish, we use the fact that $\mathcal{K}$ is
Kurepa. Fix $A\in\mathcal{K}$. Because $\mathcal{K}$ is Kurepa, there is a
countable $\mathcal{F}\subseteq\mathcal{K}$ such that $\left\\{A\cap B\colon
B\in\mathcal{K}\right\\}=\left\\{A\cap B\colon B\in\mathcal{F}\right\\}$. This
implies

$\left\\{A\cap B\colon B\in\mathcal{K}\text{ and }B\prec
A\right\\}=\left\\{A\cap B\colon B\in\mathcal{G}\right\\}$

for some (countable) $\mathcal{G}\subseteq\mathcal{F}$. In particular,

$\displaystyle Y_{A}$ $\displaystyle=\textstyle
A^{\omega}\setminus\bigcup\left\\{B^{\omega}\colon B\prec A\right\\}$
$\displaystyle=\textstyle A^{\omega}\setminus\bigcup\left\\{(B\cap
A)^{\omega}\colon B\prec A\right\\}$ $\displaystyle=\textstyle
A^{\omega}\setminus\bigcup\left\\{(B\cap A)^{\omega}\colon
B\in\mathcal{G}\right\\}$ $\displaystyle=\textstyle
A^{\omega}\setminus\bigcup\left\\{B^{\omega}\colon B\in\mathcal{G}\right\\}.$

Because $A^{\omega}$ is second countable and closed, and each of the
$B^{\omega}$ is also closed, this shows $Y_{A}$ is second countable and
$G_{\delta}$, i.e., Polish. ∎

######  Corollary 4.5.

If $0<n<\omega$, then $\mathfrak{par}(\omega_{n}^{\omega})=\aleph_{n}$.
Furthermore, if $0^{\dagger}$ does not exist then for every cardinal
$\kappa>\aleph_{0}$,

$\mathfrak{par}(\kappa^{\omega})=\mathrm{cf}[\kappa]^{\omega}=\begin{cases}\kappa&\text{if
}\mathrm{cf}(\kappa)>\omega,\\\ \kappa^{+}&\text{if
}\mathrm{cf}(\kappa)=\omega.\end{cases}$

###### Proof.

This follows from Corollary 2.6 and Theorems 2.1 and 4.4. ∎

######  Theorem 4.6.

Assume $0^{\dagger}$ does not exist, and let $\kappa$ be a singular cardinal
with $\mathrm{cf}(\kappa)=\omega$. If
$\kappa\in\mathfrak{sp}(\text{\emph{\small Borel}})$, then
$\kappa^{+}\in\mathfrak{sp}(\text{\emph{\small Borel}})$ also.

###### Proof.

This follows immediately from Theorem 4.3 and Corollary 4.5. ∎

This completes the second proof of our main theorem. To end this section, we
point out how the results from this section relate to what is sometimes called
the _Banach problem_ , posed by Stefan Banach in the Scottish Book:

  * When does a metric space admit a continuous bijective mapping onto a compact metric space?

######  Theorem 4.7.

Let $\kappa$ be an uncountable cardinal with
$\mathfrak{par}(\kappa^{\omega})=\kappa$. Then the following are equivalent:

  1. (1)

$\kappa\in\mathfrak{sp}(\text{\emph{\small Borel}})$.

  2. (2)

There is a continuous bijective mapping from $\kappa^{\omega}$ onto
$\mathbb{R}$.

  3. (3)

There is a continuous bijective mapping from $\kappa^{\omega}$ onto a compact
metric space.

###### Proof.

$(1)\Rightarrow(2)$ was proved in Lemma 4.2. $(2)\Rightarrow(3)$ because there
is a continuous bijection from $\mathbb{R}$ to a compact metric space: for
example, it is easy to describe a continuous bijection from $\mathbb{R}$ to a
wedge sum of two circles (an “8”). Finally, $(3)\Rightarrow(1)$ by the proof
of Lemma 4.1, as the role of $\mathbb{R}$ in that proof can be played equally
well by any compact metric space. ∎

By Theorem 4.4, the hypothesis of this theorem is satisfied by
$\kappa=\aleph_{n}$ for any $n<\omega$; or if $0^{\dagger}$ does not exist, it
is satisfied by any cardinal $\kappa$ with $\mathrm{cf}(\kappa)>\omega$.

A special case of this theorem (for $\kappa<\aleph_{\omega}$) was proved
already in [25] and [5]. In [25, Theorem 2.1], it is shown that the three
equivalent conditions of Theorem 4.7 also imply that if $X$ is any Banach
space of weight $\kappa$, then there is a continuous bijection from $X$ to the
Hilbert cube $[0,1]^{\omega}$.

The implication $(1)\Rightarrow(2)$ in Theorem 4.7 holds for any $\kappa$. But
the results of this section tell us nothing about whether the converse may
hold for singular cardinals of countable cofinality. (It is proved in [3,
Section 2] that if $\mathrm{cf}(\kappa)=\omega$ then
$\mathfrak{par}(\kappa^{\omega})\geq\mathrm{cf}[\kappa]^{\omega}\geq\kappa^{+}$.)

######  Question 4.8.

Suppose $\lambda$ is a singular cardinal with $\mathrm{cf}(\lambda)=\omega$,
and there is a continuous bijection $\lambda^{\omega}\to\mathbb{R}$. Does this
imply $\lambda\in\mathfrak{sp}(\text{\emph{\small Borel}})$?

######  Question 4.9.

In particular, is it consistent that there is a continuous bijection
$\omega_{\omega}^{\omega}\to\mathbb{R}$, but that
$\aleph_{\omega}\notin\mathfrak{sp}(\text{\emph{\small Borel}})$?

##  5\. Other partition spectra

Given a pointclass $\Gamma$ of sets, define

$\mathfrak{sp}({\text{\footnotesize$\Gamma$}})=\left\\{|\mathcal{P}|\colon\mathcal{P}\text{
is a partition of }\mathbb{R}\text{ into uncountably many sets in
}\Gamma\right\\}\\!.$

Many of the known results concerning $\mathfrak{sp}(\text{\small Borel})$
listed in the introduction apply equally well to other pointclasses of sets.
For example, both $\mathfrak{sp}(\text{\small closed})$ and
$\mathfrak{sp}(\text{\footnotesize OD($\mathbb{R}$)})$ satisfy the analogues
of statements $(2)$-$(6)$ in the introduction [4, 1]. They also both satisfy
the conclusion of the main result of the present paper. For
$\mathfrak{sp}(\text{\footnotesize OD($\mathbb{R}$)})$ this is relatively easy
to see: the proof given in Section 3 generalizes readily to any pointclass
$\Gamma$ that is closed under taking countable intersections and relative
complements.

######  Theorem 5.1.

Suppose $\Gamma$ is a pointclass closed under taking countable intersections
and relative complements. Assuming $0^{\dagger}$ does not exist, if $\kappa$
is a singular cardinal with $\mathrm{cf}(\kappa)=\omega$ and
$\kappa\in\mathfrak{sp}({\text{\footnotesize$\Gamma$}})$, then
$\kappa^{+}\in\mathfrak{sp}({\text{\footnotesize$\Gamma$}})$.

We now show that the conclusion of this theorem applies to several familiar
pointclasses that are not closed under taking countable intersections and
relative complements.

######  Theorem 5.2.

$\mathfrak{sp}(\text{\emph{\small
Borel}})=\mathfrak{sp}(\text{\emph{\small$\mathbf{\Sigma}^{1}_{1}$}})=\mathfrak{sp}(\text{\emph{\small$\mathbf{\Pi}^{1}_{1}$}})=\mathfrak{sp}(\text{\emph{\small$\mathbf{\Sigma}^{1}_{2}$}})=\mathfrak{sp}(\text{\emph{\small$\aleph_{1}$-Suslin}})$.

###### Proof.

Let $\Gamma$ be any of the latter four pointclasses in the statement of the
theorem. Every set in $\Gamma$ can be be written as a union of $\aleph_{1}$
Borel sets (see [17, Chapter 13]). This implies every set in $\Gamma$ can be
partitioned into $\leq\\!\aleph_{1}$ Borel sets. Thus any partition
$\mathcal{P}$ of $\mathbb{R}$ into sets in $\Gamma$ can be refined to a
partition of $\mathbb{R}$ into Borel sets, by replacing each set in
$\mathcal{P}$ with $\leq\\!\aleph_{1}$ Borel sets; and if $\mathcal{P}$ is
uncountable, any such refinement has size $|\mathcal{P}|$. Thus
$\mathfrak{sp}({\text{\footnotesize$\Gamma$}})\subseteq\mathfrak{sp}(\text{\small
Borel})$. The reverse inclusion is immediate: every Borel set is in $\Gamma$,
so a partition of $\mathbb{R}$ into Borel sets is already a partition of
$\mathbb{R}$ into sets in $\Gamma$. ∎

From this and Theorem 3.3 it follows that the conclusion of Theorem 5.1
applies when $\Gamma\in\\{\text{analytic, coanalytic,
$\mathbf{\Sigma}^{1}_{2}$, $\aleph_{1}$-Suslin}\\}$.

Next we show that the same is true for $\Gamma={}$closed. It is known that,
consistently, $\mathfrak{sp}(\text{\small
closed})\neq\mathfrak{sp}(\text{\small Borel})$. This result can be attributed
to Sierpiński [27], who showed (in our terminology) that
$\min\\!\left(\vphantom{f^{2}}\mathfrak{sp}(\text{\small
closed})\right)\geq\mathbf{cov}(\mathcal{M})$, which means we need not have
$\aleph_{1}\in\mathfrak{sp}(\text{\small closed})$. Nonetheless, we show here
that $\mathfrak{sp}(\text{\small closed})$ is a final segment of
$\mathfrak{sp}(\text{\small Borel})$, which is enough to guarantee that
Theorem 5.1 holds for $\Gamma={}$closed.

Recall that the cardinal invariant $\mathfrak{a}_{T}$ denotes the least size
of a partition of the Baire space $\omega^{\omega}$ into compact sets. (The
notation “$\mathfrak{a}_{T}$” derives from the fact that $\mathfrak{a}_{T}$ is
equal to the least size of a maximal infinite family of almost disjoint
subtrees of $2^{<\omega}$.) It is known that
$\mathfrak{d}\leq\mathfrak{a}_{T}$, and that this inequality is consistently
strict [28].

######  Lemma 5.3.

For any uncountable Polish space $X$,

$\displaystyle\mathfrak{sp}(\text{\emph{\small closed}})$
$\displaystyle=\left\\{\kappa>\aleph_{0}\colon\text{there is a partition of
}X\text{ into }\kappa\text{ compact sets}\right\\}$
$\displaystyle=\left\\{\kappa>\aleph_{0}\colon\text{there is a partition of
}X\text{ into }\kappa\text{ closed sets}\right\\}$
$\displaystyle=\left\\{\kappa>\aleph_{0}\colon\text{there is a partition of
}X\text{ into }\kappa\ F_{\sigma}\text{ sets}\right\\}.$

###### Proof.

See [4, Corollary 2.5]. ∎

######  Lemma 5.4.

$\mathfrak{sp}(\text{\emph{\small compact}})=\mathfrak{sp}(\text{\emph{\small
closed}})=\mathfrak{sp}(\text{\emph{\small$F_{\sigma}$}})=[\mathfrak{a}_{T},\mathfrak{c}]\cap\mathfrak{sp}(\text{\emph{\small
Borel}})$.

###### Proof.

By Lemma 5.3, it suffices to show $\mathfrak{sp}(\text{\small
compact})=[\mathfrak{a}_{T},\mathfrak{c}]\cap\mathfrak{sp}(\text{\small
Borel})$.

By Lemma 5.3 and the definition of $\mathfrak{a}_{T}$,
$\mathfrak{sp}(\text{\small
compact})\subseteq[\mathfrak{a}_{T},\mathfrak{c}]$. And clearly every
partition of $\mathbb{R}$ into compact sets is also a partition into Borel
sets, so $\mathfrak{sp}(\text{\small
compact})\subseteq\mathfrak{sp}(\text{\small Borel})$. Hence
$\mathfrak{sp}(\text{\small
compact})\subseteq[\mathfrak{a}_{T},\mathfrak{c}]\cap\mathfrak{sp}(\text{\small
Borel})$.

Now suppose
$\kappa\in[\mathfrak{a}_{T},\mathfrak{c}]\cap\mathfrak{sp}(\text{\small
Borel})$, and let $\mathcal{P}$ be a partition of $\mathbb{R}$ into $\kappa$
Borel sets. By a theorem of Lusin and Souslin [18, Theorem 13.7], every Borel
subset of a Polish space is the bijective continuous image of a closed subset
of $\omega^{\omega}$. In particular, for every $A\in\mathcal{P}$, there is a
continuous bijection $f_{A}:F_{A}\to A$, where $F_{A}$ is a closed subset of
$\omega^{\omega}$. Every closed subset of $\omega^{\omega}$ is Polish, so for
each $A\in\mathcal{P}$ there are two possibilities: (1) $A$ is countable, thus
so is $F_{A}$, or (2) $A$ and $F_{A}$ are uncountable, and (by Lemma 5.3)
there is a partition of $F_{A}$ into $\mathfrak{a}_{T}$ compact sets. Let
$\mathcal{P}_{1}=\left\\{A\in\mathcal{P}\colon A\text{ is countable}\right\\}$
and $\mathcal{P}_{2}=\mathcal{P}\setminus\mathcal{P}_{1}$. For each
$A\in\mathcal{P}_{2}$, fix a partition $\mathcal{Q}_{A}$ of $A$ into
$\mathfrak{a}_{T}$ compact sets. Finally, observe that

$\left\\{\\{x\\}\colon x\in A\text{ for some
}A\in\mathcal{P}_{1}\right\\}\cup\left\\{f_{A}[X]\colon
A\in\mathcal{P}_{2}\text{ and }X\in\mathcal{Q}_{A}\right\\}$

is a partition of $\mathbb{R}$ into $\kappa$ compact sets. ∎

######  Corollary 5.5.

Assume $0^{\dagger}$ does not exist. if $\kappa$ is a singular cardinal with
$\mathrm{cf}(\kappa)=\omega$ and $\kappa\in\mathfrak{sp}(\text{\emph{\small
closed}})$, then $\kappa^{+}\in\mathfrak{sp}(\text{\emph{\small closed}})$.

In addition to $\mathfrak{sp}(\text{\small closed})$,
$\mathfrak{sp}(\text{\small Borel})$, and $\mathfrak{sp}(\text{\footnotesize
OD($\mathbb{R}$)})$, which have been studied extensively, one may also find in
the literature some interest in
$\mathfrak{sp}({\text{\footnotesize$G_{\delta}$}})$. Most notably, Shelah and
Fremlin prove in [10] that
$\min\\!\left(\vphantom{f^{2}}\mathfrak{sp}({\text{\footnotesize$G_{\delta}$}})\right)\geq\mathbf{cov}(\mathcal{M}$)
(which implies, in particular, there may be no partition of $\mathbb{R}$ into
$\aleph_{1}$ $G_{\delta}$ sets). This raises the intriguing possibility that
an analogue of Lemma 5.4 might hold for
$\mathfrak{sp}({\text{\footnotesize$G_{\delta}$}})$.

######  Question 5.6.

Is
$\mathbf{cov}(\mathcal{M})=\min\\!\left(\vphantom{f^{2}}\mathfrak{sp}(\text{\emph{\footnotesize$G_{\delta}$}})\right)$?

######  Question 5.7.

Is $\mathfrak{sp}(\text{\emph{\footnotesize$G_{\delta}$}})$ a final segment of
$\mathfrak{sp}(\text{\small Borel})$?

######  Question 5.8.

If the answer to Question 5.7 is negative, is it nonetheless the case that
Theorem 5.1 holds with $\Gamma=G_{\delta}$?

## References

  * [1] A. R. Blass, “Simple cardinal characteristics of the continuum,” in Haim Judah (ed.) _Set Theory of the Reals_ , vol. 6 of Israel Mathematical Conferences Proceedings, pp. 63–90. American Mathematical Society, Provindence (1993).
  * [2] W. R. Brian, “Completely ultrametrizable spaces and continuous bijections,” _Topology Proceedings_ 45 (2015), pp. 233–252.
  * [3] W. R. Brian, “Covering versus partitioning with Polish spaces,” to appear in _Fundamenta Mathematicae_ , currently available at https://arxiv.org/abs/2101.10088.
  * [4] W. R. Brian, “Partitioning the real line into Borel sets,” submitted for publication, currently available at https://arxiv.org/abs/2112.00535.
  * [5] W. Brian and A. W. Miller, “Partitions of $2^{\omega}$ and completely ultrametrizable spaces,” _Topology and its Applications_ 184 (2015), pp. 61–71.
  * [6] A. Dodd and R. B. Jensen, “The covering lemma for $K$,” _Annals of Mathematical Logic_ 22 (1982), pp. 1–30.
  * [7] A. Dodd and R. B. Jensen, “The covering lemma for $L[U]$,” _Annals of Mathematical Logic_ 22 (1982), pp. 127–135.
  * [8] R. Engelking, _General Topology._ Sigma Series in Pure Mathematics, 6, Heldermann, Berlin (revised edition), 1989.
  * [9] M. Eskew and Y. Haiyut, “On the consistency of local and global versions of Chang’s Conjecture,” _Transactions of the American Mathematical Society_ 370 (2018), pp. 2879–2905.
  * [10] D. Fremlin and S. Shelah, “On partitions of the real line,” _Israel Journal of Mathematics_ 32 (1979), pp. 299–304.
  * [11] M. Gitik, “The negation of the singular cardinal hypothesis from $o(\kappa)=\kappa^{++}$,” _Annals of Pure and Applied Logic_ 43 (1989), pp. 209–234.
  * [12] M. Gitik, “The strength of the failure of the singular cardinal hypothesis,” _Annals of Pure and Applied Logic_ 51 (1991), pp. 215–240.
  * [13] F. Hausdorff, “Summen von $\aleph_{1}$ Mengen,” _Fundamenta Mathematicae_ 26(1) (1936), pp. 241–255.
  * [14] S. Hechler, “Short complete nested sequences in $\beta\mathbb{N}\setminus\mathbb{N}$ and small maximal almost-disjoint families,” _General Topology and Its Applications_ 2 (1972), pp. 139–149.
  * [15] R. B. Jensen and J. Steel, “$K$ without the measurable,” _Journal of Symbolic Logic_ 78 (2013), pp. 708–734.
  * [16] W. Just, A. R. D. Mathias, K. Prikry, and P. Simon, “On the existence of large $p$-ideals,” _Journal of Symbolic Logic_ 55 (1990), pp. 457–465.
  * [17] A. Kanamori, _The Higher Infinite_ ($2^{\mathrm{nd}}$ edition), Springer Monographs in Mathematics, Springer-Verlag, 2003.
  * [18] A. Kechris, _Classical Descriptive Set Theory_ , Graduate Texts in Mathematics, vol. 156, Springer-Verlag, 1995.
  * [19] M. Kojman, D. Milovich, and S. Spadaro, “Noetherian type in topological products,” _Israel Journal of Mathematics_ 202 (2014), pp. 195–225.
  * [20] J. P. Levinski, M. Magidor, and S. Shelah, “Chang’s conjecture for $\aleph_{\omega}$,” _Israel Journal of Mathematics_ 69 (1990), pp. 161–172.
  * [21] P. Matet, “Large cardinals and covering numbers,” _Fundamenta Mathematicae_ 205 (2009), pp. 45–75.
  * [22] A. W. Miller, “Covering $2^{\omega}$ with $\omega_{1}$ disjoint closed sets,” in _The Kleene Symposium_ , eds. J. Barwise, H. J. Keisler, and K. Kunen, North-Holland (1980), pp. 415–421.
  * [23] A. W. Miller, “Infinite combinatorics and definability,” _Annals of Pure and Applied Logic_ 41 (1989), pp. 179–203.
  * [24] W. Mitchell, “The Covering Lemma,” in _The Handbook of Set Theory_ , M. Foreman and A. Kanamori (eds.), Springer, 2010, pp. 1497–1594.
  * [25] A. Osipov, “Note on the Banach Problem 1 of condensations of Banach spaces onto compacta,” preprint currently available at https://arxiv.org/abs/2112.10718.
  * [26] E. Schimmerling and M. Zeeman, “Square in core models,” _Bulletin of Symbolic Logic_ 7 (2001), pp. 305–314.
  * [27] W. Sierpiński, “Un théorème sur les continus,” T$\hat{\mathrm{o}}$hoku Mathematical Journal 13 (1918), pp. 300–303.
  * [28] O. Spinas, “Partition numbers,” _Annals of Pure and Applied Logic_ 90 (1997), pp. 243–262.
  * [29] S. Todorčević, _Walks on Ordinals_ , Progress in Mathematics vol. 263. Birkhäuser Verlag, Basel, 2007.