product topology and subspace topology

Let $X_{\alpha }$ with $\alpha \in A$ be a collection of topological spaces,and let $Z_{\alpha }\subseteq X_{\alpha }$ be subsets. Let

$$X=\prod _{\alpha }{X}_{\alpha }$$

and

$$Z=\prod _{\alpha }{Z}_{\alpha }.$$

In other words, $z\in Z$ means that $z$ is a function$z:A\to {\cup }_{\alpha }{Z}_{\alpha }$such that $z(\alpha )\in Z_{\alpha }$ for each $\alpha$. Thus, $z\in X$ andwe have

$$Z\subseteq X$$

as sets.

Theorem 1.

The product topology of $Z$ coincides with the subspace topology induced by $X$.

Proof.

Let us denote by ${\tau }_X$ and ${\tau }_Z$ the product topologies for $X$ and $Z$, respectively.Also, let

$${\pi }_{X,\alpha }:X\to X_{\alpha },{\pi }_{Z,\alpha }:Z\to Z_{\alpha }$$

be the canonical projections defined for $X$ and $Z$.The subbases (http://planetmath.org/Subbasis) for $X$ and $Z$ are given by

${\beta }_X$$=$$\mathrm{\{}{\pi }_{X,\alpha }^{-1}(U):\alpha \in A,U\in \tau (X_{\alpha })\},$

${\beta }_Z$$=$$\mathrm{\{}{\pi }_{Z,\alpha }^{-1}(U):\alpha \in A,U\in \tau (Z_{\alpha })\},$

where $\tau (X_{\alpha })$ is the topology of $X_{\alpha }$ and $\tau (Z_{\alpha })$ is thesubspace topology of $Z_{\alpha }\subseteq X_{\alpha }$.The claim follows as

$${\beta }_Z=\{B\cap Z:B\in {\beta }_X\}.$$

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