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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.
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    General suggestion: when things get as abstract as $\alpha\in \Lambda$, try writing $i\in \{1,2\}$ instead. The problem becomes: show that the map $f:X_1\to X_1\times X_2$ defined by $f(x)=(x,p)$ with fixed $p$ is a homeomorphism.2012-12-25

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