$\{M \in \mathcal{P}(\Bbb Z): \:\: M=\emptyset \: \textrm{or } (-13\in M \wedge13 \not \in M )\}$ give characterisation of compact, non-finite sets

Question

Let $\mathcal T_{\Bbb Z}$ be the following topology on $\Bbb Z$:

$$\mathcal T_{\Bbb Z}:=\{M \in \mathcal{P}(\Bbb Z): \:\:\: M=\emptyset\: \: \textrm{or } M=\Bbb Z \: \: \textrm{or } (-13\in M \wedge13 \not \in M )\}$$ Now i have to give "a characterisation of all non-finite, compact spaces of $(\Bbb Z,\mathcal T)$".

I wasn't really sure what the task meant by "characterisation". I think i have to give the propertys that a non-finite sets in $(\Bbb Z,\mathcal T)$ has to have in order to be compact, but i don't really know how to do this.

Any ideas or tipps? Thanks in advance.

Answer 1

Hint: Let $A\subseteq \Bbb Z$. Show that $\mathcal U=\bigl\{\,\{-13,a\}\mid a\in A\,\bigr\}$ is an open cover of $A$ unless $13\in A$. And if $13\in A$, what can you say about any open cover?