 Solution

Write $Z={\coprod }_{n\in \mathbb{N}}{X}_n$, where each $X_n$ is a labelled copy of the topological space $X$, with its topological disjoint union topology. Define the bijectionFor a subset $U\subseteq \mathbb{N}\times X$, let $U_n=\{x\in X:(n,x)\in U\}$. By definition of the topological disjoint union, $F^{-1}(U)$ is open in $Z$ exactly when each $U_n$ is open in $X$.

The same condition characterizes open sets in the product topology on $\mathbb{N}\times X$. If $U$ is open, the map $x\mapsto (n,x)$ is continuous, so each $U_n$ is open. Conversely, if each $U_n$ is open, thenis open, since the discrete topology makes every $\{n\}$ open in $\mathbb{N}$. Hence $F$ and its inverse function are continuous, proving the required homeomorphism. This also covers the case $X=\varnothing$.