that $z^{+m} < x < z^{+(m+1)}$ (note that $x$ can not equal any of the $z^{+n}$ since $x\notin {\mathscr{Z}}$). But since the sequence $\{z_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ converges to $x$, the open neighbourhood $(z^{+m},z^{+(m+1)})$ of $x$ contains infinitely many entries of the sequence $\{z_i\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$. Since the $z_i$ are in ${\mathscr{Z}}$, this clearly contradicts the definition of $z^{+(m+1)}$.

The Igusa–Todorov cluster categories ${\mathscr{C}}({\mathscr{Z}})$ of Dynkin type $A_{ \infty }$ {#sec:IT}
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In the rest of the paper, $k$ is an algebraically closed field.

Igusa and Todorov [@IT:cyclicposets] constructed a cluster category $\cC( \cZ )$. They proved in [@IT:cyclicposets sec. 2.4] that it has the following properties.

1.  $\cC( \cZ )$ is a $k$-linear Hom-finite Krull–Schmidt triangulated category.

2.  $\cC( \cZ )$ is $2$-Calabi–Yau, that is, there are natural isomorphisms $$\Ext_{ \cC( \cZ ) }^1(X,Y) \cong \mathrm{D}\Ext_{ \cC( \cZ ) }^1(Y,X)$$ where $\mathrm{D}(-) = \Hom_k( -,k )$.

3.  If $X = \{ x_0,x_1 \}$ is a diagonal of $\cZ$, then there is an indecomposable object $E( X ) = E( x_0,x_1 )$ in $\cC( \cZ )$, and this induces a bijection from diagonals of $\cZ$ to isomorphism classes of indecomposable objects of $\cC( \cZ )$.

4.  The suspension functor 