# Two characterizations of topological spaces with no infinite discrete
subspace

Jean Goubault-Larrecq  LSV, ENS Paris-Saclay, CNRS, Université Paris-Saclay,
61 av. président Wilson, 94235 Cachan Cedex<EMAIL_ADDRESS>and  Maurice
Pouzet  Univ. Lyon, Université Claude-Bernard Lyon1, CNRS UMR 5208, Institut
Camille Jordan, 43 bd. 11 Novembre 1918, 69622 Villeurbanne Cedex, France and
Mathematics & Statistics Department, University of Calgary, Calgary, Alberta,
Canada T2N 1N4<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We give two characteristic properties of topological spaces with no infinite
discrete subspaces. The first one was obtained recently by the first author.
The full result extends well-known characterizations of posets with no
infinite antichain.

###### Key words and phrases:

Ordered sets, Closure systems, Noetherian topological spaces, Well-quasi-
orders

###### 1991 Mathematics Subject Classification:

Primary

The first author was supported by grant ANR-17-CE40-0028 of the French
National Research Agency ANR (project BRAVAS)

A topological space $T:=(E,\mathcal{F})$, where $\mathcal{F}$ is the set of
closed subsets, is _noetherian_ if every descending sequence of closed
subspaces is stationary. A subset $X$ of $E$ is _discrete_ if every subset of
$X$ is closed with respect to the induced topology. A closed subset is
_irreducible_ if it is non-empty and not the union of two proper closed
subsets.

######  Theorem 1.

The following properties are equivalent for a topological space
$T:=(E,\mathcal{F})$.

  1. (i)

No infinite subset of $E$ is discrete;

  2. (ii)

Every closed set is a finite union of irreducible closed subsets;

  3. (iii)

Every closed set contains a dense subset on which the induced topology is
noetherian.

The equivalence between $(i)$ and $(ii)$ was proved recently by the first
author [9]. We give a proof of the equivalence of $(iii)$ with $(i)$ and
$(ii)$ in Section 1. That also gives an alternative proof of the equivalence
between $(i)$ and $(ii)$, and will stress the importance of the notion of
infinite separating chain of closed sets, inspired from [3, 2]. We make
additional remarks in Section 2.

##  1\. The proof

We mimic the proof of a similar result for posets, fairly well-known, which we
give at the end of the paper. The significant part is the implication
$(i)\Rightarrow(iii)$.

We recall that a _closure system_ is a pair $(E,\varphi)$ where $\varphi$ (the
_closure_) is a map from the power set of $E$ into itself which is extensive,
order-preserving and idempotent. A subset $C\subseteq E$ is _closed_ if
$\varphi(C)=C$; it is _independent_ if $x\not\in\varphi(C\setminus\\{x\\})$
for every $x\in C$; it is _generating_ if $\varphi(C)=E$. The closure
$\varphi_{\restriction E^{\prime}}$ _induced by $\varphi$_ on a subset
$E^{\prime}$ of $E$ is defined by $\varphi_{\restriction
E^{\prime}}(X):=\varphi(X)\cap E^{\prime}$ for every $X\subseteq E^{\prime}$.
We recall that $\varphi$ defines a topology if and only if it preserves finite
unions. In this case, we say that $\varphi$ is _topological_ , also
independent sets are called _discrete_ , generating sets are called _dense_.

We will use the following notion and result, adapted from [3, Section 3, p.7],
see also [2, p.25]. A non-empty descending chain $\mathcal{I}$ of closed sets
of $(E,\varphi)$ is separating if for every
$I\in\mathcal{I}\setminus\\{\bigcup\mathcal{I}\\}$ and every finite set
$F\subseteq\bigcup\mathcal{I}\setminus I$, there is a set
$J\in\mathcal{I}$—necessarily included in $I$—such that
$I\not\subseteq\varphi(F\cup J)$.

######  Lemma 1.

A closure system $(E,\varphi)$ contains an infinite independent set if and
only if $E$ contains a subset $E^{\prime}$ for which the induced closure
contains an infinite separating chain of closed sets.

###### Proof.

Any infinite independent set contains a countably infinite independent subset,
so we may as well assume a given infinite independent set $X$ of the form
$\\{x_{n}:n<\omega\\}$, where $x_{m}\neq x_{n}$ for all $m\neq n$. Set
$E^{\prime}:=X$. Then the chain $\mathcal{I}=\\{I_{n}:n<\omega\\}$, where
$I_{n}:=\varphi_{\restriction E^{\prime}}(X\setminus\\{x_{i}:i<n\\})$, is
separating in $E^{\prime}$. Indeed, first
$\bigcup\mathcal{I}=I_{0}=E^{\prime}=X$, next every
$I\in\mathcal{I}\setminus\\{\bigcup\mathcal{I}\\}$ is an $I_{n}$ for some
$n\geq 1$. For every finite set $F$ of points of $I_{0}\setminus I_{n}$,
define $J$ as $I_{n+1}$. Since $x_{n}$ is different from every $x_{i}$, $i<n$,
$x_{n}$ is in $X\setminus\\{x_{i}:i<n\\}$ hence in $I_{n}$. It follows that
$x_{n}$ is not in $F$. It is not in $J$ either, because
$J\subseteq\varphi(X\setminus\\{x_{i}:i<n+1\\})\subseteq\varphi(X\setminus\\{x_{n}\\})$,
and $X$ is independent. Therefore $x_{n}$ is not in $F\cup J$. We rewrite that
as $F\cup J\subseteq X\setminus\\{x_{n}\\}$, and conclude that $\varphi(F\cup
J)\subseteq\varphi(X\setminus\\{x_{n}\\})$. Since $X$ is independent again,
$x_{n}$ cannot be in $\varphi(F\cup J)$. However, $x_{n}$ is in $I_{n}$, so
$I_{n}\not\subseteq\varphi(F\cup J)$.

Conversely, let $E^{\prime}$ be a subset of $E$ such that the induced closure
$\varphi^{\prime}:=\varphi_{\restriction E^{\prime}}$ on $E^{\prime}$ contains
an infinite separating chain $\mathcal{I}$ of closed sets. We construct an
infinite independent subset for the induced closure $\varphi^{\prime}$. It
will be independent for the closure $\varphi$. For that, define inductively an
infinite sequence $x_{0},I_{0},\ldots,x_{n},I_{n},\ldots$ such that
$I_{0}\in\mathcal{I}\setminus\\{\bigcup\mathcal{I}\\},x_{0}\in\bigcup\mathcal{I}\setminus
I_{0}$ and such that, for every $n\geq 1$:

  1. $(a_{n})$

$I_{n}\in\mathcal{I}$;

  2. $(b_{n})$

$I_{n}\subset I_{n-1}$;

  3. $(c_{n})$

$x_{n}\in I_{n-1}\setminus\varphi^{\prime}(\\{x_{0},\ldots,x_{n-1}\\}\cup
I_{n})$.

Since $\mathcal{I}$ is infinite,
$\mathcal{I}\setminus\\{\bigcup\mathcal{I}\\}\not=\emptyset$. Choose
arbitrarily $I_{0}\in\mathcal{I}\setminus\\{\bigcup\mathcal{I}\\}$ and
$x_{0}\in\bigcup\mathcal{I}\setminus I_{0}$. Let $n\geq 1$. Suppose
$x_{k},I_{k}$ are defined and satisfy $(a_{k})$, $(b_{k})$, $(c_{k})$ for
every $k\leq n-1$. Set $I:=I_{n-1}$ and $F:=\\{x_{0},\ldots,x_{n-1}\\}$. Since
$I\in\mathcal{I}$ and $F\subseteq\bigcup\mathcal{I}\setminus I$, there is some
$J\in\mathcal{I}$ such that $I\not\subseteq\varphi^{\prime}(F\cup J)$. Let
$z\in I\setminus\varphi^{\prime}(F\cup J)$. Set $x_{n}:=z$, $I_{n}:=J$.

It remains to check that the set $X:=\\{x_{n}:n<\omega\\}$ is independent. For
every $x\in X$, say $x=x_{n}$, we know that $x_{n}$ is in $I_{n-1}$ and not in
the closed set $C:=\varphi^{\prime}(\\{x_{0},\ldots,x_{n-1}\\}\cup I_{n})$, by
$(c_{n})$. The set $C$ contains $x_{0}$, …, $x_{n-1}$. For every $k>n+1$,
$I_{k-1}\subset I_{n}$ by $(b_{k-1})$, …, $(b_{n+1})$, and $x_{k}\in I_{k-1}$
by $(c_{k})$, so $I_{n}$, hence also $C$, contains every $x_{k}$ with $k>n$.
It follows that $C$ contains every element of $X$ except $x=x_{n}$. Therefore
$C\supseteq\varphi^{\prime}(X\setminus\\{x\\})$, from which we conclude that
$x$, not being in $C$, is not in $\varphi^{\prime}(X\setminus\\{x\\})$ either.
∎

######  Remark 1.

Assuming $\varphi$ topological, we can take $E^{\prime}=E$ in Lemma 1. Indeed,
let $X$ be an independent subset of the form $\\{x_{n}:n<\omega\\}$. Let
$I_{n}:=\varphi(X\setminus\\{x_{i}:i<n\\})$. The chain
$\mathcal{I}=\\{I_{n}:n<\omega\\}$ is separating, as we now check. First
$\bigcup\mathcal{I}=I_{0}=\varphi(X)$. Every
$I\in\mathcal{I}\setminus\\{\bigcup\mathcal{I}\\}$ is an $I_{n}$ for some
$n\geq 1$. For every finite set $F$ of points of $\varphi(X)\setminus I_{n}$,
we define $J$ as $I_{n+1}$ and we check that $I_{n}\not\subseteq\varphi(F\cup
J)$ by showing that $x_{n}$, which is in $I_{n}$, is not in $\varphi(F\cup
I_{n+1})$. Since $\varphi$ is topological, it suffices to show that $x_{n}$ is
neither in $\varphi(F)$ nor in $\varphi(I_{n+1})=I_{n+1}$. The latter—that
$x_{n}$ is not in $I_{n+1}$—is clear. As for the former, we note that
$I_{n}\cup\varphi(X\setminus\\{x_{n}\\})=\varphi((X\setminus\\{x_{i}:i<n\\})\cup(X\setminus\\{x_{n}\\}))=\varphi(X)$,
using the fact that $\varphi$ is topological. That implies
$\varphi(X)\setminus I_{n}\subseteq\varphi(X\setminus\\{x_{n}\\})$, so
$F\subseteq\varphi(X\setminus\\{x_{n}\\})$, whence
$\varphi(F)\subseteq\varphi(X\setminus\\{x_{n}\\})$. If $x_{n}$ were in
$\varphi(F)$, it would then be in $\varphi(X\setminus\\{x_{n}\\})$, and that
is impossible since $X$ is independent.

######  Remark 2.

In general, e.g. if $\varphi$ is not topological, we cannot take
$E=E^{\prime}$.

Consider $E:=\powerset(\mathbb{N})$,
$\varphi(\mathcal{A}):=\powerset(\bigcup\mathcal{A})$. Note that the closed
subsets of $E$ are exactly the sets of the form $\powerset(A)$ with
$A\subseteq\mathbb{N}$. There is an infinite independent set, say
$X:=\\{\\{n\\}\mid n\in\mathbb{N}\\}$. However, no subset of $E$ has any
separating chain of closed sets $\mathcal{I}$. Indeed, assume one existed.
Pick $I\in\mathcal{I}\setminus\\{\bigcup\mathcal{I}\\}$. Since $I$ is closed,
$I=\powerset(A)$ for some $A\subseteq\mathbb{N}$. Take $F:=\\{A\\}$: for every
$J\in\mathcal{I}$, $\varphi(F\cup J)\supseteq\varphi(F)=\powerset(A)=I$, so
$\mathcal{I}$ cannot be separating.

In a quasi-ordered set $E$, for every $A\subseteq E$, write
$\mathop{\downarrow}\nolimits A$ for $\\{x\in E:\exists y\in A,x\leq y\\}$. A
subset $A$ is _cofinal_ in $E$ if and only if $E=\mathop{\downarrow}\nolimits
A$. An _initial segment_ is a subset $I$ of $E$ such that
$I=\mathop{\downarrow}\nolimits I$. The _finitely generated_ initial segments
are the sets of the form $\mathop{\downarrow}\nolimits A$, $A$ finite. We
write $\mathop{\downarrow}\nolimits x$ for
$\mathop{\downarrow}\nolimits\\{x\\}$.

Every closure system $E$ is quasi-ordered by $x\leq y$ if and only if
$x\in\varphi(\\{y\\})$. For every $x\in X$, $\mathop{\downarrow}\nolimits
x=\varphi(\\{x\\})$. Every closed set is an initial segment. If $\varphi$ is
topological, then every finitely generated initial segment
$\mathop{\downarrow}\nolimits A$ is equal to $\varphi(A)$, hence is closed.

We say that a quasi-ordered set is _well-founded_ if and only if it has no
infinite strictly descending sequence $x_{0}>x_{1}>\cdots>x_{n}>\cdots$, where
$x<y$ if and only if $x\leq y$ and $y\not\leq x$. That extends the same notion
on posets.

######  Lemma 2.

If $(E,\varphi)$ is a closure system then $E$ contains a generating subset $D$
on which the collection of finitely generated initial segments of the quasi-
ordered set $(D,\leq_{\restriction D})$ is well-founded.

###### Proof.

According to a result of Birkhoff (1937), the poset $\mathbf{I}_{<\omega}(P)$
of finitely generated initial segments of a poset $P$ is well-founded if $P$
is well-founded [1, Theorem 2, p.182]. This property holds for initial
segments of a quasi-ordered set too, since initial segments of a quasi-ordered
set are inverse images of initial segments of the order quotient. For the
reader’s convenience, we give a proof of Birkhoff’s result. Let
$\mathop{\downarrow}\nolimits A_{1}\supset\mathop{\downarrow}\nolimits
A_{2}\supset\cdots\supset\mathop{\downarrow}\nolimits A_{n}\supset$ be an
infinite descending sequence where each $A_{i}$ is finite. We may assume that
each $A_{i}$ is an antichain. Construct a tree $T$ whose vertices are finite
chains $\\{x_{1},x_{2},\cdots,x_{n}\\}$ where each $x_{i}\in A_{i}$ and
$x_{1}\geq x_{2}\geq\cdots\geq x_{n}$. The cardinality of such a set is $n$,
or some lower number (if some element appears several times), and is the
_depth_ of the vertex in $T$. The unique parent of a depth $n$ set, $n\geq 1$,
is obtained by removing its least element ($x_{n}$, but also $x_{n-1}$ if that
happens to be equal to $x_{n}$, and so on). The empty chain is the root. Every
element of each $A_{n}$ is the least element of at least one such chain, of
depth at most $n$. Hence every set $\mathop{\downarrow}\nolimits A_{n}$
appears as the initial segment generated by a finite set of vertices of $T$.
Since there are infinitely many sets $\mathop{\downarrow}\nolimits A_{n}$, the
tree $T$ is infinite. Since $T$ is finitely branching, by Kőnig’s Lemma it has
an infinite branch, and that is an infinite descending sequence of elements:
contradiction.

Thus, in order to prove the lemma, it suffices to prove that $E$ contains a
generating subset on which the quasi-order $\leq$ is well-founded.

A result due to Hausdorff (see [5, Chapter 2, p.57]) states that every poset
contains a well-founded cofinal subset. That is also valid for quasi-ordered
sets such as $E$. Indeed, consider the family $\mathcal{W}$ of all well-
founded subsets of the set $E$, and order it by prefix: $A\sqsubseteq B$ if
and only if $B\cap\mathop{\downarrow}\nolimits A=A$. By Zorn’s Lemma, it has a
maximal element $A$. If $A$ were not cofinal, there would be a point $x$ that
is not in $\mathop{\downarrow}\nolimits A$. Then $B=A\cup\\{x\\}$ would be a
strictly larger well-founded subset of $E$, contradicting maximality.

Hence let $D$ be a well-founded cofinal subset of $E$. For every $x\in E$,
there is a point $y\in D$ such that $x\leq y$. In other words, $x$ is in
$\varphi(\\{y\\})$, hence in the larger set $\varphi(D)$. Therefore
$\varphi(D)=E$ and $D$ is generating. ∎

### Proof of the implication $(i)\Rightarrow(iii)$.

Let $\varphi$ be the closure associated with $\mathcal{F}$, and remember that
it is topological. Let $C$ be a closed set. We define a dense subset of $C$ on
which the induced topology is noetherian. This will be $D$, as given by Lemma
2. Note that $D$ is dense in $C$.

Let $\varphi_{\restriction D}$ be the closure induced on $D$, namely
$\varphi_{\restriction D}(X):=\varphi(X)\cap D$ for every $X\subseteq D$. This
is also a topological closure, and we claim that it is noetherian.

By $(i)$, $E$ contains no infinite discrete set, so $D$ does not contain any
infinite discrete set either. Imagine that $D$ contained an infinite strictly
descending sequence $I_{0}\supset I_{1}\supset\cdots\supset
I_{n}\supset\cdots$ of closed subsets. By Lemma 1, that chain must fail to be
separating: there must be an index $m_{1}\geq 1$ and a finite set $F_{1}$ of
points of $I_{0}\setminus I_{m_{1}}$ such that, for every $n<\omega$ (in
particular, for every $n>m_{1}$), $I_{m_{1}}\subseteq\varphi_{\restriction
D}(F_{1}\cup I_{n})=\varphi_{\restriction D}(F_{1})\cup I_{n}$. The last
equality is because $\varphi_{\restriction D}$ is topological, and $I_{n}$ is
closed. Hence $m_{1}\geq 1$, $F_{1}\subseteq I_{0}\setminus I_{m_{1}}$, and
for every $n>m_{1}$, $I_{m_{1}}\setminus I_{n}\subseteq\varphi_{\restriction
D}(F_{1})$. We do the same with the infinite subsequence starting at
$I_{m_{1}}$: there is an index $m_{2}\geq m_{1}+1$ and a finite set
$F_{2}\subseteq I_{m_{1}}\setminus I_{m_{2}}$ such that for every $n>m_{2}$,
$I_{m_{2}}\setminus I_{n}\subseteq\varphi_{\restriction D}(F_{2})$. Proceeding
this way, we obtain indices $m_{k+1}\geq m_{k}+1$ and finite sets
$F_{k+1}\subseteq I_{m_{k}}\setminus I_{m_{k+1}}$ such that for every
$n>m_{k+1}$, $I_{m_{k+1}}\setminus I_{n}\subseteq\varphi_{\restriction
D}(F_{k+1})$, for every $k\geq 1$.

Note that $F_{k+1}\subseteq I_{m_{k}}\setminus
I_{m_{k+1}}\subseteq\varphi_{\restriction D}(F_{k})$, so
$\varphi_{\restriction D}(F_{k+1})\subseteq\varphi_{\restriction D}(F_{k})$.
It follows that the sequence ${(\varphi_{\restriction D}(F_{k}))}_{k\geq 1}$
is descending. Since $D$ was obtained from Lemma 2, that sequence must be
finite. Pick $k\geq 2$ such that $\varphi_{\restriction
D}(F_{k})=\varphi_{\restriction D}(F_{k+1})$. $F_{k}$ cannot be empty, since
$\varphi_{\restriction D}(F_{k})$ contains $I_{m_{k}}\setminus I_{m_{k+1}}$,
which is non-empty. Pick $x\in F_{k}$. In particular, $x$ is in
$I_{m_{k-1}}\setminus I_{m_{k}}$, hence is not in $I_{m_{k}}$. However, $x$ is
also in $\varphi_{\restriction D}(F_{k})=\varphi_{\restriction D}(F_{k+1})$,
and since $F_{k+1}\subseteq I_{m_{k}}\setminus I_{m_{k+1}}\subseteq
I_{m_{k}}$, $x$ is also in $\varphi_{\restriction D}(I_{m_{k}})=I_{m_{k}}$:
contradiction.

### Proof of the implication $(iii)\Rightarrow(ii)$.

Let $\varphi$ be the closure on $E$. Let $C$ be a closed set and $D$ be a
dense subset of $C$ on which the closure $\varphi_{\restriction D}$ is well-
founded.

On $D$ every closed set $D^{\prime}$ is a finite union of irreducible closed
sets. This fact goes back to Noether, see [1, Chapter VIII, Corollary, p.181].
Indeed, if $D$ is not such, then, since the collection of closed sets on $D$
is well-founded, there is a minimal member $C^{\prime}$ which is not a finite
union of irreducible members. In particular, $C^{\prime}$ is non-empty. If
$C^{\prime}$ is the union of two proper closed subsets, by minimality those
closed subsets must be finite unions of irreducible subsets of $D$, hence so
must be $C^{\prime}$. It follows that $C^{\prime}$ is irreducible:
contradiction.

Now $D$ is itself closed in $D$, so we can write $D$ as a finite union of
irreducible closed subsets $C_{i}$ of $D$, $1\leq i\leq n$. For each $C_{i}$,
$\varphi(C_{i})$ is irreducible in $E$, as one easily checks [8, Lemma
8.4.10]. By density and the fact that $\varphi$ is topological,
$C=\varphi(D)=\bigcup_{i=1}^{n}\varphi(C_{i})$.

### Proof of the implication $(ii)\Rightarrow(i)$

Let $\varphi$ be the closure on $E$ again, and let $X$ be a discrete subspace.
Write $\varphi(X)$ as a finite union of irreducible closed sets $I_{1}$, …,
$I_{n}$.

For each $x\in X$, $x$ is in some $I_{k}$. We claim that
$I_{k}=\varphi(\\{x\\})$. To that end, we note that
$I_{k}\subseteq\varphi(X)=\varphi(\\{x\\})\cup\varphi(X\setminus\\{x\\})$,
since $\varphi$ is topological. Therefore $I_{k}$ is equal to the union of the
two closed sets $\varphi(\\{x\\})\cap I_{k}$ and
$\varphi(X\setminus\\{x\\})\cap I_{k}$. Since $X$ is discrete, hence
independent, $x$ is not in $\varphi(X\setminus\\{x\\})$, and as $x\in I_{k}$,
$\varphi(X\setminus\\{x\\})\cap I_{k}$ is a proper closed subset of $I_{k}$.
Because $I_{k}$ is irreducible, $\varphi(\\{x\\})\cap I_{k}$ cannot be a
proper closed subset of $I_{k}$, so $\varphi(\\{x\\})\cap I_{k}=I_{k}$. This
means that $I_{k}\subseteq\varphi(\\{x\\})$, and the converse inclusion
follows from $x\in I_{k}$.

It follows that for any two distinct points $x,y\in X$, $x$ and $y$ cannot be
in the same $I_{k}$. Otherwise $\varphi(\\{x\\})=\varphi(\\{y\\})$, but since
$X$ is independent, $x$ is not in $\varphi(X\setminus\\{x\\})$, hence not in
the smaller set $\varphi(\\{y\\})$. That is impossible since
$\varphi(\\{x\\})=\varphi(\\{y\\})$ contains $x$.

Since each $I_{k}$ can contain at most one point from $X$, $X$ is finite.

##  2\. Remarks and comments

###  2.1. Other characterizations

A.H. Stone [14, Theorem 2] has shown that $(i)$ is equivalent to two further
properties: $(iv)$ every open cover of every subspace $X$ of $T$ has a finite
subfamily whose union is dense in $X$, and $(v)$ every continuous real-valued
function on every subspace of $T$ is bounded.

###  2.2. Noetherian topological spaces

Noetherian topological spaces have been studied for their own sake by A.H.
Stone [14]. They are an important basic notion in algebraic geometry, since
the spectrum of any noetherian ring in a noetherian topological space, with
the so-called Zariski topology. They have also found applications in
verification, the domain of computer science concerned with finding algorithms
that prove properties of other computer systems, automatically [7]. One can
consult Section 9.7 of [8], which is devoted to noetherian topological spaces.

###  2.3. A related result

The implication $(i)\Rightarrow(ii)$ follows from the following result about
closure systems. We recall that an _up-directed_ subset of a poset $P$ is a
non-empty subset $A$ of $P$ such that any two elements of $A$ have an upper
bound in $A$, and that an _ideal_ is an up-directed initial segment. We always
order powersets by inclusion.

######  Theorem 2.

[6] If a closure system $(E,\varphi)$ contains no infinite independent set
then: $(*)$ there are finitely many pairwise disjoint subsets $A_{i}$ of $E$
and, for each $A_{i}$, a proper ideal $N_{i}$ of $\powerset(A_{i})$ such that
for every $X\subseteq\bigcup_{i\in I}A_{i}$, the set $X$ generates $E$ if and
only if $A_{i}\cap X\not\in N_{i}$, for each $i\in I$.

As it will become apparent in Proposition 3, this result specialized to
topological closures is just implication $(i)\Rightarrow(ii)$. Decompositions
of topological closures were considered in [6], but this consequence was
totally missed.

######  Remark 3.

In Theorem 2, we may suppose that:

(1) $A_{i}\cap\varphi(\bigcup_{j\in
I\setminus\\{i\\}}A_{i})=\emptyset\;\text{for each}\;i\in I.$

This is Remark 1 of [6]. We repeat the argument. If $(\ref{equ:key})$ does not
hold, set $A^{\prime}_{i}:=A_{i}\setminus\varphi(\bigcup_{j\in
I\setminus\\{i\\}}A_{i})$ and
$N^{\prime}_{i}:=N_{i}\cap\powerset(A^{\prime}_{i})$. Let us check that the
$A^{\prime}_{i}$ and $N^{\prime}_{i}$ satisfy the condition in Theorem 2.
First, each $N^{\prime}_{i}$ is a proper ideal (if $N^{\prime}_{i}$ is not
proper, that is $A^{\prime}_{i}\in N^{\prime}_{i}$, then $A^{\prime}_{i}\in
N_{i}$. Set $X:=A^{\prime}_{i}\cup\bigcup_{j\not=i}A_{j}$; since
$A_{i}\subseteq A^{\prime}_{i}\cup\varphi(X)$, $X$ generates $E$, but violates
the condition of the theorem). Next, let
$X\subseteq\bigcup\\{A^{\prime}_{i}:i\in I\\}$. Suppose that $X$ generates
$E$. Since $X\subseteq\bigcup\\{A_{i}:i\in I\\}$, $A_{i}\cap X\not\in N_{i}$,
for each $i\in I$. Hence $A^{\prime}_{i}\cap X=A_{i}\cap X\not\in
N^{\prime}_{i}$. Conversely, if $A^{\prime}_{i}\cap X\not\in N^{\prime}_{i}$,
for each $i\in I$, then $A^{\prime}_{i}\not\in N_{i}$, hence $X$ generates
$E$.

######  Proposition 3.

Let $\varphi$ be a topological closure operator on a set $E$. Then $E$ is a
finite union of irreducible closed sets iff $E$ has a decomposition satisfying
the condition on generating sets in Theorem 2.

###### Proof.

The result is a consequence of the following two claims.

######  Claim 4.

Let $A_{i}$ ($i\in I$) be a finite decomposition satisfying Condition $(*)$ on
generating sets of Theorem 2. According to Remark 3, we may assume that it
satisfies Condition (1). Then $Y\in N_{i}$ iff $Y\subseteq A_{i}$ and
$\varphi(Y)\not\supseteq A_{i}$. In particular, $X_{i}:=\varphi(A_{i})$ is
irreducible.

###### Proof of Claim 4..

First note that $E=\varphi(\bigcup_{j\in I}A_{j})$. Indeed, it suffices to
apply $(*)$ to $X:=\bigcup_{j\in I}A_{j}$, and to realize that $A_{i}\cap
X=A_{i}$ cannot be in $N_{i}$, since $N_{i}$ is a proper ideal of
$\powerset(A_{i})$.

Suppose that $Y\in N_{i}$. Since $N_{i}$ is an initial segment, $A_{i}\cap Y$
is in $N_{i}$. For $X:=Y\cup\bigcup_{j\in I\setminus\\{i\\}}A_{j}$, $A_{i}\cap
X=A_{i}\cap Y$ by Condition (1), so $A_{i}\cap X$ is in $N_{i}$. By $(*)$, $X$
does not generate $E$. Since $\varphi$ is a closure operator, if
$A_{i}\subseteq\varphi(Y)$ then $E=\varphi(\bigcup_{j\in
I}A_{j})\subseteq\varphi(Y\cup\bigcup_{j\in
I\setminus\\{i\\}}A_{j})=\varphi(X)$, which is impossible. Hence
$A_{i}\not\subseteq\varphi(Y)$.

Conversely, since $\varphi(Y)\not\supseteq A_{i}$, there is a point $x$ in
$A_{i}$—hence not in $\varphi(\bigcup_{j\in I\setminus\\{i\\}}A_{i})$ by
Condition (1)—which is not in $\varphi(Y)$, hence not in
$\varphi(Y)\cup\varphi(\bigcup_{j\in I\setminus\\{i\\}}A_{j})$. The latter is
equal to $\varphi(X)$, where $X:=Y\cup\bigcup_{j\in I\setminus\\{i\\}}A_{j}$,
since $\varphi$ is topological, so $X$ does not generate $E$. Using $(*)$,
$A_{j}\cap X$ is in $N_{j}$ for some $j\in I$. If $j\neq i$, then $A_{j}\cap
X\supseteq A_{j}$, and $A_{j}\cap X\in N_{j}$ would imply $A_{j}\in N_{j}$,
contradicting the fact that $N_{j}$ is proper. Therefore $j=i$. This means
that $A_{i}\cap X$, which is equal to $A_{i}\cap Y$ by Condition (1), hence to
$Y$ since $Y\subseteq A_{i}$, is in $N_{i}$.

We finally show that $X_{i}$ is irreducible. Since $N_{i}$ is proper, $A_{i}$
is non-empty, hence $X_{i}$ is non-empty. Assume that $X_{i}$ is the union of
two proper closed subsets $C_{1}$ and $C_{2}$. Consider $Y:=C_{1}\cap A_{i}$
(resp., $Y:=C_{2}\cap A_{i}$). Then $Y\subseteq A_{i}$ and
$\varphi(Y)\subseteq C_{1}$ (resp., $C_{2}$) cannot contain
$X_{i}=\varphi(A_{i})$, hence cannot contain $A_{i}$. It follows that $Y$ is
in $N_{i}$. In other words, both $C_{1}\cap A_{i}$ and $C_{2}\cap A_{i}$ are
in $N_{i}$. Since $N_{i}$ is an ideal, $(C_{1}\cap A_{i})\cup(C_{2}\cap
A_{i})=(C_{1}\cup C_{2})\cap A_{i}=X_{i}\cap A_{i}=A_{i}$ is in $N_{i}$, which
is impossible since $N_{i}$ is proper. ∎

######  Claim 5.

If $E$ is a finite union of irreducible closed sets, let $(X_{i})_{i\in I}$ be
a family of such sets with $|I|$ minimum. Set
$A_{i}:=X_{i}\setminus\bigcup_{j\not=i}X_{j}$ and
$N_{i}:=\\{A^{\prime}\subseteq A_{i}:\varphi(A^{\prime})\not=X_{i}\\}$. This
decomposition satisfies Condition $(*)$ of Theorem 2.

###### Proof of Claim 5..

We check that $N_{i}$ is an ideal. Given $A^{\prime},B^{\prime}\in N_{i}$,
$\varphi(A^{\prime}\cup
B^{\prime})=\varphi(A^{\prime})\cup\varphi(B^{\prime})$, since $\varphi$ is
topological. If that were equal to the whole of $X_{i}$, and since
$\varphi(A^{\prime})$ and $\varphi(B^{\prime})$ are both proper closed subsets
of $X_{i}$, $X_{i}$ would fail to be irreducible. Hence
$\varphi(A^{\prime}\cup B^{\prime})\neq X_{i}$, so that $A^{\prime}\cup
B^{\prime}$ is in $N_{i}$.

Then we check that $N_{i}$ is proper, namely that $A_{i}$ is not in $N_{i}$.
By definition of $A_{i}$, $X_{i}\subseteq
A_{i}\cup(\bigcup_{j\not=i}X_{j})\subseteq\varphi(A_{i})\cup(\bigcup_{j\not=i}X_{j})$,
so
$X_{i}=(X_{i}\cap\varphi(A_{i}))\cup(X_{i}\cap(\bigcup_{j\not=i}X_{j}))=\varphi(A_{i})\cup(X_{i}\cap(\bigcup_{j\not=i}X_{j}))$,
a union of two closed subsets. The second one,
$X_{i}\cap(\bigcup_{j\not=i}X_{j})$, is a proper subset of $X_{i}$ since we
have chosen a family of least cardinality. Since $X_{i}$ is irreducible, the
other one cannot be a proper subset. Therefore $\varphi(A_{i})=X_{i}$. It
follows that $A_{i}$ is not in $N_{i}$.

Finally, let $X\subseteq\bigcup_{i\in I}A_{i}$.

If $A_{i}\cap X$ belongs to $N_{i}$ for no $i\in I$, then by definition of
$N_{i}$, $\varphi(A_{i}\cap X)=X_{i}$, hence $\varphi(X)=\varphi(\bigcup_{i\in
I}A_{i}\cap X)=\bigcup_{i\in I}\varphi(A_{i}\cap X)=\bigcup_{i\in I}X_{i}=E$.

Conversely, assume that $\varphi(X)=E$. Since $X\subseteq\bigcup_{j\in
I}A_{j}$, $X=\bigcup_{j\in I}A_{j}\cap X$, and since $\varphi$ is topological,
$E=\varphi(X)=\bigcup_{j\in I}\varphi(A_{j}\cap X)$. Recall that
$E=\bigcup_{i\in I}X_{i}$, so for every $i\in I$, $X_{i}\subseteq\bigcup_{j\in
I}\varphi(A_{j}\cap X)$, and since $X_{i}$ is irreducible, there is a $j\in I$
such that $X_{i}\subseteq\varphi(A_{j}\cap X)$. If $j\neq i$, then
$X_{i}\subseteq\varphi(A_{j})\subseteq\varphi(X_{j})=X_{j}$, which is
impossible since we have chosen ${(X_{i})}_{i\in I}$ of least cardinality.
Hence $j=i$, meaning that for every $i\in I$, $X_{i}\subseteq\varphi(A_{i}\cap
X)$. Since $\varphi(A_{i}\cap
X)\subseteq\varphi(A_{i})\subseteq\varphi(X_{i})=X_{i}$,
$X_{i}=\varphi(A_{i}\cap X)$, and that shows that $A_{i}\cap X$ is not in
$N_{i}$. ∎

###  2.4. Topological properties versus lattice properties

Item $(ii)$ in Theorem 1 is a property about the lattice of closed sets of a
topological space: if two topological spaces have the same lattice of closed
sets, they both satisfy $ii)$ or none satisfies it. The same is true for item
$(i)$. Indeed, as it is well-known, the existence of an infinite discrete
subspace (or more generally of an infinite independent subset for a closure
system) amounts to the existence of an embedding of $\powerset(\mathbb{N})$,
the collection of subsets of $\mathbb{N}$ ordered by inclusion, into the
lattice of closed sets. It not clear that this is the case for item $(iii)$
without having a proof of Theorem 1. For example, the notion of density is not
a lattice property: the smallest cardinality of dense subsets of $\mathbb{N}$
with the cofinite topology (whose closed sets are the finite subsets of
$\mathbb{N}$ plus $\mathbb{N}$ itself) is infinite, but the sobrification
$\mathbb{N}^{s}$ of this space obtained by adding a new point $\infty$ to
$\mathbb{N}$, and whose closed sets are the finite subsets of $\mathbb{N}$
plus $\mathbb{N}^{s}$, has a one-point dense subset, $\\{\infty\\}$, but an
isomorphic lattice of closed sets.

###  2.5. The case of posets

Theorem 1 has a well-known predecessor in the theory of poset. It is worth to
recall it.

Let $P$ be a poset, and $A$ be a subset of $P$. An _upper bound_ of $A$ is any
$z\in P$ such that $x\leq z$ for every $x\in A$. The set $A$ is _up-
independent_ if no pair of distinct members of $A$ have a common upper bound;
it is _consistent_ (or compatible) if every pair of members of $A$ has an
upper bound.

The _final segments_ of $P$ are the initial segments of $P^{d}$, the opposite
order; we denote by $\mathop{\uparrow}\nolimits A$, resp.
$\mathop{\uparrow}\nolimits a$, the final segment of $P$ generated by
$A\subseteq P$, resp. $a\in P$.

The set $\mathbf{I}(P)$ of initial segments of $P$ is the set of closed sets
of a topology, the Alexandroff topology. In this setting, a subset $A$ is
discrete if and only if it is an antichain, $A$ is dense if and only if it is
cofinal, and $A$ is irreducible if and only if it is an ideal.

A poset $P$ is _well-quasi-ordered_ (w.q.o. for short) if it is well-founded
and contains no infinite antichain. According to Higman [11], $P$ is w.q.o.
iff $\mathbf{I}(P)$ is well-founded.

We recall the following result (see [5, Chapter 4]):

######  Theorem 6.

The following properties are equivalent for a poset $P$:

  1. (a)

$P$ contains no infinite antichain;

  2. (b)

every initial segment of $P$ is a finite union of ideals;

  3. (c)

every initial segment of $P$ contains a cofinal subset which is well-quasi-
ordered.

###### Proof.

This is just Theorem 1 applied to $P$ with the Alexandroff topology, provided
one notes that a poset is well-quasi-ordered if and only if it is noetherian
in its Alexandroff topology. But the proof simplifies.

$(a)\Rightarrow(c)$. Let $P^{\prime}$ be an initial segment. By an already
cited result of Hausdorff, $P^{\prime}$ contains a well-founded cofinal subset
$A$. Since $P$ has no infinite antichain, $P^{\prime}$ has no infinite
antichain; being well-founded it is w.q.o.

$(c)\Rightarrow(b)$. Let $P^{\prime}$ be an initial segment and $A$ be a
cofinal subset of $P^{\prime}$ which is w.q.o. Being w.q.o., $A$ is a finite
union of ideals $I_{1},\dots,I_{q}$. This is a basic result of the theory of
w.q.o. [1, Chapter VIII, Corollary, p.181]. Indeed, as in the proof of
implication $(iii)\Rightarrow(ii)$ of Theorem 1, replacing “closed” by
“initial segment” and “irreducible closed” by “ideal”, if $A$ is not such,
then, since $\mathbf{I}(A)$ is well-founded, there is a minimal member
$A^{\prime}\in\mathbf{I}(A)$ which is not a finite union of ideals. This
$A^{\prime}$ is irreducible, hence is an ideal: contradiction.

Now $P^{\prime}=\mathop{\downarrow}\nolimits A=\mathop{\downarrow}\nolimits
I_{1}\cup\dots\mathop{\downarrow}\nolimits I_{q}$ and the set
$\mathop{\downarrow}\nolimits I_{i}$ are ideals of $P^{\prime}$.

$(b)\Rightarrow(a)$. Let $A$ be an antichain of $P$. An ideal $I$ of
$\mathop{\downarrow}\nolimits A$ cannot contain more than one element of $A$.
Since $\mathop{\downarrow}\nolimits A$ is a finite union of ideals, $A$ is
finite. ∎

######  Remark 4.

_A direct proof of $(a)\Rightarrow(b)$ can be obtained from a special case of
a result of Erdős and Tarski [4]. This special case is a prototypal Min-Max
result which has been overlooked. For the reader’s convenience, we recall it
and give a proof._

######  Proposition 7.

If a poset $P$ contains no infinite up-independent set then there is a finite
upper bound on the size of up-independent sets. In this case, the maximum size
of up-independent sets, the least number of ideals whose union is $P$ and the
least number of consistent sets whose union is $P$ are equal.

###### Proof.

Let $P$ be a poset. Let $Q:=\\{x\in P:\mathop{\uparrow}\nolimits x\text{ is
up-directed}\\}$, let $U:=\mathop{\downarrow}\nolimits Q$ and $R:=P\setminus
U$.

If $R$ is non-empty, define recursively a map $f$ from $2^{<\omega}$, the set
of finite $0$-$1$ sequences, into $R$, by picking some arbitrary element of
$R$ for $f(\epsilon)$ (where $\epsilon$ is the empty sequence), and choosing
two incompatible elements of $\mathop{\uparrow}\nolimits f(s)$ for $f(s.0)$
and $f(s.1)$ (here $s.0$ and $s.1$ are the two sequences obtained from $s$ by
adding $0$ and $1$ to the right of the sequence $s$, respectively). Ordered by
extension of sequences, $2^{<\omega}$ is the dyadic tree $T_{2}$; the map $f$
from $T_{2}$ preserves the incompatibility relation. Since $T_{2}$ contains
infinite up-independent sets, so does $R$. Hence $R$ is empty.

Among the subsets of $Q$ which are up-independent (in $Q$) let $L$ be a
maximal member with respect to inclusion and $\kappa:=|L|$. Then
$\mathop{\uparrow}\nolimits L$ is cofinal in $Q$ hence in $U$. Since for each
$x\in L$, the set $I_{x}:=\mathop{\downarrow}\nolimits F_{x}$, where
$F_{x}:=\mathop{\uparrow}\nolimits x$, is an ideal, $U$ is covered by at most
$\kappa$ ideals. Since $R$ is empty, the poset $P$ is covered by at most
$\kappa$, and in fact exactly $\kappa$, ideals. It is immediate that $\kappa$
is the common value to the parameters defined in the above proposition. ∎

Now, assuming $(a)$, $(b)$ can be proved as follows. Let $I$ be an initial
segment of $P$. Since $P$ contains no infinite antichain, $I$ does not contain
an infinite antichain either. In the subposet $I$, the up-independent subsets
are all finite, since every up-independent subset is an antichain. Their
maximal size is then the least number of ideals whose union is $I$, by the
Erdős-Tarski theorem.

## References

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#  12th Workshop on the CKM Unitarity Triangle
Santiago de Compostela, 18-22 September 2023
Measurements of the CKM angle $\gamma$ and parameters related to mixing and
$C\\!P$ violation in the charm at LHCb

Innes Mackay1 on behalf of the LHCb Collaboration
1University of Oxford Oxford  United Kingdom

###### Abstract

A recent combination of measurements performed by the LHCb collaboration
determined that $\gamma=(63.8^{+3.5}_{-3.7})^{\circ}$. The fit combined the
results of $\gamma$ and charm measurements, which resulted in precision
improvements for the strong-phase difference between
${{\mathrm{D}}^{0}}\to{{\mathrm{K}}^{-}}{{\pi}^{+}}$ and
${{\mathrm{D}}^{0}}\to{{\mathrm{K}}^{+}}{{\pi}^{-}}$ decays,
$\delta_{{{\mathrm{K}}}\pi}^{\mathrm{D}}$, and the ${\mathrm{D}}^{0}$–${\kern
1.00006pt\overline{\kern-1.00006pt\mathrm{D}}}{}^{0}$ mixing parameter, $y$.
In addition, new LHCb measurements of the CKM angle $\gamma$ using
${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}$ and
${{{\mathrm{B}}^{-}}}\to{{\mathrm{D}}^{*}}{{\mathrm{K}}^{-}}$ decays with the
${\mathrm{D}}\to{{\mathrm{K}}^{0}_{\mathrm{S}}}{\mathrm{h}}^{+}{\mathrm{h}}^{-}$
final state (where ${\mathrm{h}}=\pi,{{\mathrm{K}}}$) are presented.

##  1 LHCb combination of $\gamma$ and charm measurements

A key goal of flavour physics is to test the unitarity of the CKM matrix by
overconstraining the Unitarity Triangle (UT). The angle $\gamma$ is
particularly interesting because it can be measured with a negligible
theoretical uncertainty using tree-level decays [1]. The direct measurements
of $\gamma$ are compared to indirect determinations from global fits to
parameters related to the other angles and sides of the UT [2, 3] that are
measured in decays with additional loop diagrams. Discrepancies between the
direct and indirect measurements of $\gamma$ would be a sign of beyond
Standard Model effects.

The angle $\gamma$ is most precisely measured using
${{{\mathrm{B}}^{-}}}\to{\mathrm{D}}{{\mathrm{K}}^{-}}$ decays, where
$\mathrm{D}$ is a superposition of ${\mathrm{D}}^{0}$ and ${\kern
1.00006pt\overline{\kern-1.00006pt\mathrm{D}}}{}^{0}$ mesons. The ratio of
amplitudes for the two decay paths is

$r_{\mathrm{B}}e^{i(\delta_{\mathrm{B}}-\gamma)}=\frac{\rm{A}({{{\mathrm{B}}^{-}}}\to{{\kern
1.00006pt\overline{\kern-1.00006pt\mathrm{D}}}{}^{0}}{{\mathrm{K}}^{-}})}{\rm{A}({{{\mathrm{B}}^{-}}}\to{{\mathrm{D}}^{0}}{{\mathrm{K}}^{-}})},$
(1.1)

where $r_{\mathrm{B}}$ and $\delta_{\mathrm{B}}$ are the amplitude ratio and
strong-phase difference between the decays. The angle $\gamma$ can be
determined by exploiting interference between
${{{\mathrm{B}}^{-}}}\to{{\mathrm{D}}^{0}}(\to\mathrm{f}){{\mathrm{K}}^{-}}$
and ${{{\mathrm{B}}^{-}}}\to{{\kern
1.00006pt\overline{\kern-1.00006pt\mathrm{D}}}{}^{0}}(\to\mathrm{f}){{\mathrm{K}}^{-}}$
decays, where $\mathrm{f}$ is a common final state to both ${\mathrm{D}}^{0}$
and ${\kern 1.00006pt\overline{\kern-1.00006pt\mathrm{D}}}{}^{0}$ mesons. The
same equations above apply to
${{{\mathrm{B}}^{+}}}\to{\mathrm{D}}{{\mathrm{K}}^{+}}$ decays under the
exchange $\gamma\to-\gamma$. The procedure can be generalised to
${\mathrm{B}}^{0}$ and ${\mathrm{B}}^{0}_{\rm{s}}$ decays, because the
sensitivity to $\gamma$ originates from the interference between
${\rm{b}}\to{\rm{u}}$ and ${\rm{b}}\to{\rm{c}}$ quark transitions that is
shared by all $\mathrm{B}$-meson decays into a mixture of ${\mathrm{D}}^{0}$
and ${\kern 1.00006pt\overline{\kern-1.00006pt\mathrm{D}}}{}^{0}$ decays.

The squared amplitude for the
${{{\mathrm{B}}^{-}}}\to{\mathrm{D}}(\to\mathrm{f}){{\mathrm{K}}^{-}}$ decay
is

$|\rm{A}({{{\mathrm{B}}^{-}}})|^{2}\propto\rm{A}_{\mathrm{D}}^{2}+\rm{r}_{\mathrm{B}}^{2}\rm{A}_{{\kern
0.70004pt\overline{\kern-0.70004pt\mathrm{D}}}}^{2}+2\rm{A}_{\mathrm{D}}\rm{A}_{{\kern
0.70004pt\overline{\kern-0.70004pt\mathrm{D}}}}\rm{r}_{\mathrm{B}}\cos(\delta_{\mathrm{B}}+\delta_{\mathrm{D}}-\gamma),$
(1.2)

where $\rm{A}_{\mathrm{D}}$ and $\rm{A}_{{\kern
0.70004pt\overline{\kern-0.70004pt\mathrm{D}}}}$ are the magnitude of the
amplitudes for the ${{\mathrm{D}}^{0}}\to\mathrm{f}$ and ${{\kern
1.00006pt\overline{\kern-1.00006pt\mathrm{D}}}{}^{0}}\to\mathrm{f}$ decays,
respectively, and $\delta_{\mathrm{D}}$ is the strong-phase difference between
them. Therefore, external knowledge of the $\mathrm{D}$ decay amplitudes is
used to extract $\gamma$. The specific inputs depend on the final state of the
$\mathrm{D}$ decay, and are most often expressed in terms of the amplitude
ratio, $r_{\mathrm{D}}$, and strong-phase difference, $\delta_{\mathrm{D}}$,
between the ${{\mathrm{D}}^{0}}\to\mathrm{f}$ and ${{\kern
1.00006pt\overline{\kern-1.00006pt\mathrm{D}}}{}^{0}}\to\mathrm{f}$ decays.
The reverse scenario is also possible, such that the results of the $\gamma$
analyses can be used to constrain the values of $\delta_{\mathrm{D}}$ and
$r_{\mathrm{D}}$. This is particularly useful for the
${{\mathrm{D}}^{0}}\to{{\mathrm{K}}^{-}}{{\pi}^{+}}$ final state, because the
determination of the ${\mathrm{D}}^{0}$–${\kern
1.00006pt\overline{\kern-1.00006pt\mathrm{D}}}{}^{0}$ mixing parameters, $x$
and $y$, in this decay channel require knowledge of the strong-phase
difference $\delta_{{{\mathrm{K}}}\pi}^{\mathrm{D}}$ [4]. In 2021, the LHCb
collaboration combined the results of the $\gamma$ measurements with those in
the charm quark sector, which determined a noticeable improvement in the
precision of $\delta_{{{\mathrm{K}}}\pi}^{\mathrm{D}}$ and $y$, as is
displayed in Fig. 1 [5].

|
---|---
Figure 1: Comparison between the 1 (68.3%) to 5$\sigma$ confidence regions for
(left) ${\mathrm{r}_{{{\mathrm{K}}}\pi}^{\mathrm{D}}}$ vs.
$\delta_{{{\mathrm{K}}}\pi}^{\mathrm{D}}$ and (right) $x$ vs. $y$, as
determined in a combination of LHCb $\gamma$ and charm measurements (orange)
and charm only (blue) [5].

In the combination $\gamma$ was determined to be
$\gamma=(63.8^{+3.5}_{-3.7})^{\circ}$, which is in good agreement with the
indirect measurements $\gamma=(65.6^{+0.9}_{-2.7})^{\circ}$ [2] or
$\gamma=(65.8\pm 2.2)^{\circ}$ [3] depending on the statistical approach used.
A comparison of $\gamma$ measurements using different $\mathrm{B}$ meson
decays is displayed in Fig. 2. Those that use ${\mathrm{B}}^{\pm}$ mesons
dominate the precision and are in minor tension with measurements made using
${\mathrm{B}}^{0}_{\rm{s}}$ and ${\mathrm{B}}^{0}$ decays. However, recent
results using time-dependent ${\mathrm{B}}^{0}_{\rm{s}}$ decays [6] and time-
independent ${\mathrm{B}}^{0}$ decays [7, 8], which both use the full LHCb
dataset, will see better agreement in the next combination. Furthermore,
additional sensitivity to $\gamma$ will be achieved by including the results
of two measurements made using the
${{\mathrm{B}}^{\pm}}\to{{\mathrm{D}}^{*}}{{\mathrm{K}}^{\pm}}$ channel for
the first time [9, 10]. The remainder of these proceedings will discuss some
of the new LHCb measurements of $\gamma$ that use the
${\mathrm{D}}\to{{\mathrm{K}}^{0}_{\mathrm{S}}}{\mathrm{h}}^{+}{\mathrm{h}}^{-}$
decay final state, where ${\mathrm{h}}=\pi,{{\mathrm{K}}}$.

Figure 2: Confidence levels for values of the CKM angle $\gamma$ determined in
a combination of $\gamma$ and charm measurements at LHCb. The confidence
levels for $\gamma$ measured using (red) ${\mathrm{B}}^{\pm}$, (orange)
${\mathrm{B}}^{0}$, (light blue) ${\mathrm{B}}^{0}_{\rm{s}}$ and (dark blue)
all $\mathrm{B}$ decays are shown [5].

##  2 Measuring $\gamma$ with
${\mathrm{D}}\to{{\mathrm{K}}^{0}_{\mathrm{S}}}{\mathrm{h}}^{+}{\mathrm{h}}^{-}$
final states

The amplitude for the
${\mathrm{D}}\to{{\mathrm{K}}^{0}_{\mathrm{S}}}{\mathrm{h}}^{+}{\mathrm{h}}^{-}$
decay is phase-space dependent due to intermediate resonances. It is found
that the sensitivity to $\gamma$ is optimised by studying regions of this
phase space, which is categorised by the squared invariant masses of the
${{\mathrm{K}}^{0}_{\mathrm{S}}}{\mathrm{h}}^{+}$ and
${{\mathrm{K}}^{0}_{\mathrm{S}}}{\mathrm{h}}^{-}$ pairs, labelled $m_{+}^{2}$
and $m_{-}^{2}$, respectively [11, 12, 13, 14]. The phase space is divided
into $2\mathcal{N}$ regions labelled from $i=-\mathcal{N}$ to $i=\mathcal{N}$
(excluding 0), and is symmetrical about the line $m_{-}^{2}=m_{+}^{2}$, where
a region with $m_{-}^{2}>m_{+}^{2}$ ($m_{-}^{2}<m_{+}^{2}$) is the $i^{th}$
($-i^{th}$) bin. For the
${\mathrm{D}}\to{{\mathrm{K}}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$
(${\mathrm{D}}\to{{\mathrm{K}}^{0}_{\mathrm{S}}}{{\mathrm{K}}^{+}}{{\mathrm{K}}^{-}}$)
decay a scheme with $\mathcal{N}=8$ ($\mathcal{N}=2$) regions is used [15].

The signal yields in a phase-space region, $i$, are given by

$\displaystyle\rm{N}_{i}({{{\mathrm{B}}^{-}}}\to{\mathrm{D}}{{\mathrm{K}}^{-}})=h^{{{{\mathrm{B}}^{-}}}}\left[\rm{F}_{i}+(x_{-}^{2}+y_{-}^{2})\rm{F}_{-i}+2\sqrt{\rm{F}_{i}\rm{F}_{-i}}\left(\rm{c}_{i}x_{-}+\rm{s}_{i}y_{-}\right)\right],$
(2.1)
$\displaystyle\rm{N}_{i}({{{\mathrm{B}}^{+}}}\to{\mathrm{D}}{{\mathrm{K}}^{+}})=h^{{{{\mathrm{B}}^{+}}}}\left[\rm{F}_{-i}+(x_{+}^{2}+y_{+}^{2})\rm{F}_{i}+2\sqrt{\rm{F}_{i}\rm{F}_{-i}}\left(\rm{c}_{i}x_{+}-\rm{s}_{i}y_{+}\right)\right],$
(2.2)

where $x_{\pm}$ and $y_{\pm}$ are the $C\\!P$ violation observables defined as
$x_{\pm}=r_{\mathrm{B}}\cos(\delta_{\mathrm{B}}\pm\gamma)$ and
$y_{\pm}=r_{\mathrm{B}}\sin(\delta_{\mathrm{B}}\pm\gamma)$. Furthermore, the
$h^{{{\mathrm{B}}^{\pm}}}$ are normalisation factors which absorb the detector
effects and production asymmetries, the $\rm{F}_{i}$ parameters are the
efficiency corrected probabilities for a
${{\mathrm{D}}^{0}}\to{{\mathrm{K}}^{0}_{\mathrm{S}}}{\mathrm{h}}^{+}{\mathrm{h}}^{-}$
decay in a phase-space region, $i$, and the parameters $\rm{c}_{i}$ and
$\rm{s}_{i}$ are given by integrals over the $\mathrm{D}$ decay amplitude
averaged cosines and sines of the strong-phase difference,
$\delta_{\mathrm{D}}$,

$\displaystyle\rm{K}_{i}$
$\displaystyle=\int_{i}\mathrm{d}m_{-}^{2}\mathrm{d}m_{+}^{2}\,|\mathrm{A}_{\mathrm{D}}(m_{-}^{2},m_{+}^{2})|^{2},$
(2.3) $\displaystyle\rm{c}_{i}$
$\displaystyle=\frac{1}{\sqrt{K_{i}K_{-i}}}\int_{i}\mathrm{d}m_{-}^{2}\mathrm{d}m_{+}^{2}\,|\mathrm{A}_{\mathrm{D}}(m_{-}^{2},m_{+}^{2})||\mathrm{A}_{{\kern
0.70004pt\overline{\kern-0.70004pt\mathrm{D}}}}(m_{-}^{2},m_{+}^{2})|\cos\delta_{\mathrm{D}}(m_{-}^{2},m_{+}^{2}),$
(2.4) $\displaystyle\rm{s}_{i}$
$\displaystyle=\frac{1}{\sqrt{\rm{K}_{i}\rm{K}_{-i}}}\int_{i}\mathrm{d}m_{-}^{2}\mathrm{d}m_{+}^{2}\,|\mathrm{A}_{\mathrm{D}}(m_{-}^{2},m_{+}^{2})||\mathrm{A}_{{\kern
0.70004pt\overline{\kern-0.70004pt\mathrm{D}}}}(m_{-}^{2},m_{+}^{2})|\sin\delta_{\mathrm{D}}(m_{-}^{2},m_{+}^{2}).$
(2.5)

The $\gamma$ measurements presented in these proceedings do not rely on
amplitude models for the $\rm{c}_{i}$ and $\rm{s}_{i}$ inputs. Instead, they
are determined using quantum-correlated
${{\mathrm{D}}^{0}}{\kern-0.70007pt{{\kern
1.00006pt\overline{\kern-1.00006pt\mathrm{D}}}{}^{0}}}$ pairs produced at
charm factories [16, 17]. In each measurement, fits to the $\mathrm{B}$ and
${\kern 1.00006pt\overline{\kern-1.00006pt\mathrm{B}}}$ meson invariant mass
distributions in each phase-space region are performed with the signal yields
parameterised by equations Eqs. 2.1 and 2.2 to extract values for the $C\\!P$
violation observables and thus $\gamma$, $r_{\mathrm{B}}$ and
$\delta_{\mathrm{B}}$.

##  3 Measuring $\gamma$ in
${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}(892)$ decays

The measurement of $\gamma$ using
${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}(892)$ decays [7] is
performed without accounting for time dependence because the flavour of the
${\mathrm{B}}^{0}$ meson at decay is unambiguously tagged by the charge on the
kaon from the ${{\mathrm{K}}^{*0}}\to{{\mathrm{K}}^{+}}{{\pi}^{-}}$ decay.
Both paths to the ${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}$ decay
final state are colour suppressed, whereas only one is in
${{{\mathrm{B}}^{-}}}\to{\mathrm{D}}{{\mathrm{K}}^{-}}$ decays. Therefore, the
interference in ${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}$ decays
is expected to be around 3 times larger and thus each event has a higher
sensitivity to $\gamma$ compared to
${{{\mathrm{B}}^{-}}}\to{\mathrm{D}}{{\mathrm{K}}^{-}}$.

Figure 3: Fit to the distribution of ${{\kern
1.00006pt\overline{\kern-1.00006pt\mathrm{B}}}{}^{0}}\to{\mathrm{D}}{{\overline{{\mathrm{K}}}}^{*0}}$
candidates in the phase-space region, $i=-1$, compared to that of
${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}$ decays in the phase-
space region, $i=+1$. The asymmetry is highlighted on the plot.

Criteria are imposed to select the
${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}$ resonance in the
${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{+}}{{\pi}^{-}}$ phase space.
However, non-signal decays still remain in this region which need to be
accounted for. This is achieved by introducing a coherence factor,
$\kappa=0.958^{+0.005}_{-0.046}$ [18], which dilutes the interference terms in
Eqs. 2.1 and 2.2. In the fit to data, the $\rm{F}_{i}$ values are fixed to
those determined in ${{{\mathrm{B}}^{-}}}\to{\mathrm{D}}{{\pi}^{-}}$ decays
determined in Ref. [19]. The relative efficiency differences across the phase
space between ${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}$ and
${{{\mathrm{B}}^{-}}}\to{\mathrm{D}}{{\pi}^{-}}$ decays are examined in
simulation and found to be small. Figure 3 compares the fit to
${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}$ candidates in the
phase-space region $i=1$, with that of ${{\kern
1.00006pt\overline{\kern-1.00006pt\mathrm{B}}}{}^{0}}\to{\mathrm{D}}{{\overline{{\mathrm{K}}}}^{*0}}$
candidates in the phase-space region $i=-1$, to provide an example of the
$C\\!P$ asymmetry. The analysis determines that

$\displaystyle\gamma=(49^{+22}_{-19})^{\circ},$ $\displaystyle
r_{{\mathrm{B}}^{0}}=0.271^{+0.065}_{-0.066},$
$\displaystyle\delta_{{\mathrm{B}}^{0}}=(236^{+19}_{-21})^{\circ},$

where the uncertainties are statistically dominated. The values presented
supersede those determined using a smaller dataset in Ref. [20]. The
measurement is required to break the four fold degeneracy of $\gamma$ values
from measurements of ${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}$
decays with $\mathrm{D}$ decays to 2- and 4-body final states presented in
Ref. [8]. The combined precision is around $7^{\circ}$, with a best-fit value
that will reduce the tensions with measurements performed using
${{\mathrm{B}}^{\pm}}$ decays.

##  4 Measuring $\gamma$ using
${{\mathrm{B}}^{\pm}}\to{{\mathrm{D}}^{*}}{{\mathrm{K}}^{\pm}}$ decays

Two new measurements of $\gamma$ using
${{{\mathrm{B}}^{-}}}\to{{\mathrm{D}}^{*}}{{\mathrm{K}}^{-}}$ decays have
recently been performed by the LHCb collaboration. The study presented in Ref.
[10] reconstructs the full decay chain, whilst that described in Ref. [9] does
not reconstruct the soft neutral particle from the
${{\mathrm{D}}^{*}}\to{\mathrm{D}}X$ decay, where $X={{\pi}^{0}},\gamma$.
There are advantages and disadvantages to both measurements. In the former,
the signal yield is lower because the reconstruction efficiency of soft
neutral particles is low at LHCb; however, it has fewer backgrounds and they
are easier to distinguish.

In each measurement, ${{{\mathrm{B}}^{-}}}\to{{\mathrm{D}}^{*}}{{\pi}^{-}}$
decays are used as an additional signal channel. The interference in these
decays is small, so they don’t significantly contribute to the measurement of
$\gamma$. Instead, they are used to determine the $\rm{F}_{i}$ values which
are simultaneously fitted alongside the $C\\!P$ violation observables.

|
---|---
Figure 4: The 68.3% and 95.5% confidence regions in the $\gamma$ vs. $r_{{\mathrm{B}}^{0}}$ plane determined in the fits to the $C\\!P$ violation observables in the study of (left) fully reconstructed and (right) partially reconstructed ${{\mathrm{B}}^{\pm}}\to{{\mathrm{D}}^{*}}{{\mathrm{K}}^{\pm}}$ decays. Parameter | Fully reconstructed | Partially reconstructed
---|---|---
$\gamma$ | $(69^{+14}_{-14})^{\circ}$ | $(92^{+21}_{-17})^{\circ}$
$r_{{{\mathrm{B}}^{+}}}$ | $0.15\pm 0.03$ | $0.080^{+0.022}_{-0.023}$
$\delta_{{{\mathrm{B}}^{+}}}$ | $(311\pm 15)^{\circ}$ | $(310^{+15}_{-20})^{\circ}$
Table 1: Fitted values of $\gamma$ and the hadronic parameters determined
using fully and partially reconstructed
${{\mathrm{B}}^{\mp}}\to{{\mathrm{D}}^{*}}{{\mathrm{K}}^{\pm}}$ decays.

The values of $\gamma$, $r_{\mathrm{B}}$ and $\delta_{\mathrm{B}}$ determined
in each analysis are displayed in Tab. 1. The two measurements are found to be
statistically independent. The results are in agreement with each other, and
with the average value of $\gamma$ determined in the LHCb combination [5].
Figure 4 displays a comparison between the 68.3% and 95.5% confidence regions
in the $\gamma$ vs. $r_{{{\mathrm{B}}^{+}}}$ plane in both measurements. The
uncertainty on $\gamma$ is inversely proportional to the value of
$r_{{{\mathrm{B}}^{+}}}$ determined in the fit, which leads to larger
confidence regions in the $\gamma$ vs. $r_{{{\mathrm{B}}^{+}}}$ plane for the
partially reconstructed analysis despite the $C\\!P$ violation observables
being more precise. In combinations of measurements the partially
reconstructed analysis will have a larger statistical weight.

##  5 Summary

The LHCb collaboration continues to drive precision on the CKM angle $\gamma$,
with a current average value, determined in combination with charm
measurements, of $\gamma=(63.8^{+3.5}_{-3.7})^{\circ}$ [5]. New LHCb
measurements will be included in the next combination, and those that use the
self-conjugate
${\mathrm{D}}\to{{\mathrm{K}}^{0}_{\mathrm{S}}}{\mathrm{h}}^{+}{\mathrm{h}}^{-}$,
where ${\mathrm{h}}=\pi,{{\mathrm{K}}}$, final state are presented in these
proceedings. An analysis of
${{\mathrm{B}}^{0}}\to{\mathrm{D}}{{\mathrm{K}}^{*0}}$ decays determined that
$\gamma=(49^{+22}_{-19})^{\circ}$ [7] and will reduce tensions between
measurements made using ${{\mathrm{B}}^{\pm}}$ and ${{\mathrm{B}}^{0}}$ decays
in the next combination. Finally, two new LHCb measurements performed using
${{\mathrm{B}}^{\pm}}\to{{\mathrm{D}}^{*}}{{\mathrm{K}}^{\pm}}$ decays [9, 10]
for the first time provide additional sensitivity to $\gamma$, as is necessary
for a more precise test of CKM unitarity.

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