# 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 $z\colon A\to\cup_{\alpha}Z_{\alpha}$ such that $z(\alpha)\in Z_{\alpha}$ for each $\alpha$. Thus, $z\in X$ and we 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

 $\pi_{X,\alpha}\colon X\to X_{\alpha},\quad\pi_{Z,\alpha}\colon Z\to Z_{\alpha}$

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

 $\displaystyle\beta_{X}$ $\displaystyle=$ $\displaystyle\{\pi_{X,\alpha}^{-1}(U):\alpha\in A,U\in\tau(X_{\alpha})\},$ $\displaystyle\beta_{Z}$ $\displaystyle=$ $\displaystyle\{\pi_{Z,\alpha}^{-1}(U):\alpha\in A,U\in\tau(Z_{\alpha})\},$

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

 $\beta_{Z}=\{B\cap Z:B\in\beta_{X}\}.$

Title product topology and subspace topology ProductTopologyAndSubspaceTopology 2013-03-22 15:35:33 2013-03-22 15:35:33 matte (1858) matte (1858) 6 matte (1858) Theorem msc 54B10