= Solution
{c}
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 <bijection>
$$
\boxed{F:Z\longrightarrow\mathbb N\times X,\qquad F(x\text{ in }X_n)=(n,x).}
$$
For 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>, then
$$
U=\bigcup_{n\in\mathbb N}(\{n\}\times U_n)
$$
is <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 $X=\varnothing$.
Back to article page
