## You are here

Homeproduct topology and subspace topology

## Primary tabs

# 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 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

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

∎

## Mathematics Subject Classification

54B10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith