Showing two topological spaces are not homeomorphic 
Having infinitely many open sets in a topology is a topological property

#1: What are example of two topological spaces, preferably simple ones, which this statement show are not homeomorphic.
But, more importantly why does it show this? Having trouble understanding topological properties in connection to homeomorphisms.
#2: What about two spaces where this statement doesn't help deciding whether they are homeomorphic or not?

Take $\mathbb Z$ with the usual topology and with the topology$$\tau=\{\emptyset,\mathbb{Z}\}\cup\bigl\{\{a,a+1,a+2,\ldots\}\,|\,a\in\mathbb Z\bigr\}.$$This topology is countable, but the usual one isn't (since every subset of $\mathbb Z$ is open).
Note that if $f$ is a homeomorphism from a topological space $(X_1,\tau_1)$ onto a topological space $(X_2,\tau_2)$, then $f$ induces a bijection between $\tau_1$ and $\tau_2$:$$\begin{array}{ccc}\tau_1&\longrightarrow&\tau_2\\X&\mapsto&f(X).\end{array}$$Therefore, $\#\tau_1=\#\tau_2$.
