When $X$ is homeomorphic to $X \times \mathbb{N}$? Under what conditions $X$ is homeomorphic to $X \times \mathbb{N}$? where $\mathbb{N}$ 
is the discrete space.
 A: Let $X = \bigsqcup_{i \in \mathbb{N}} X_{i}$ be the disjoint union of countably many subspaces and each $X_{i} \approx X$, then $X \approx X \times \mathbb{N}$. On the other hand, if $X \approx X \times \mathbb{N}$ then $X$ can be partitioned into countably many clopen susbsets of $X$ each of it is homeomorphic to $X$ which is equivalent to that $X$ is the disjoint union of countably many subspaces each of it is homeomorphic to $X$. 
A: It's true that if $X\approx X\times\mathbb N$, then $X$ is the disjoint union of countably many subspaces. However, the converse isn't true! Here's a counterexample: $E=\mathbb ({-2},{-1})\cup\mathbb N$. All but one of the connected components of $E$ are singletons. The same can't be said of $E\times\mathbb N$.
Here's a more likely claim, which you may or may not consider trivial: $X\approx X\times\mathbb N$ if and only if $X\approx Y\times\mathbb N$ for some space $Y$. You can think of this as a formula that generates all spaces with the desired property. Can you prove it?
Perhaps more interesting: $X\approx X\times\mathbb N$ if and only if $X\approx A\times B$ where $A$ is a topological space and $B$ is an infinite discrete topological space.
