Let $\{(X_\alpha,\mathscr{T}_\alpha):\alpha\in\Lambda\}$ be a collection of topological spaces, and let $\mathscr{T}$ be the product topology on $X=\prod_{\alpha\in\Lambda}X_\alpha$. Let $p\in X$, let $\beta\in\Lambda$, and let $H_{p\beta}=\{x\in X;\mbox{if }\alpha\neq\beta\mbox{, then }x_\alpha=p_\alpha\}$. Define the function $f:X_\beta\to H_{p\beta}$ as follows: for each $x_\beta\in X_\beta$ let $f(x_\beta)$ be the member of $H_{p\beta}$ defined by $[f(x_\beta)]_\beta=x_\beta$ and for $\alpha\neq\beta,[f(x_\beta)]_\alpha=p_\alpha$. Then $f$ is a homeomorphism.
- I would like see a proof of this theorem please.