former result is well known and appears to be folklore; the latter is proved in [@HJR:torsionpairsA thm. A].

The aim of this paper is to prove similar key properties for the cluster categories $\cC( \cZ )$ of Dynkin type $A_{ \infty }$, which were introduced by Igusa and Todorov. Subsection \[subsec:A\] is a primer on $\cC( \cZ )$ and its combinatorial model by an $\infty$-gon, and Subsections \[subsec:B\] and \[subsec:C\] state the key properties we will prove.

Our results generalise the following parts of the literature:

-   When $\cZ$ has one, respectively two limit points (see Definition \[D:admissible\](iii)), [@HJinfinity thms. A,B,C], respectively [@LP thms. 3.13, 5.7] classified cluster tilting subcategories in $\cC( \cZ )$, and showed that they form a cluster structure in the sense of [@BIRS sec. II.1].

-   When $\cZ$ has one, respectively two limit points, [@Ng thm. 3.18], respectively [@CZZ thm. 4.4] classified torsion pairs in $\cC( \cZ )$.

Furthermore, our Theorem \[thm:B\] is closely related to [@SvR thm. 7.17].

We would also like to mention that there are a number of papers on the classification of cluster tilting subcategories and torsion pairs in more general cluster categories, mainly based on combinatorial models of Riemann surfaces with 