product topology and subspace topology


Let Xα with αA be a collectionMathworldPlanetmath of topological spacesMathworldPlanetmath, and let ZαXα be subsets. Let

X=αXα

and

Z=αZα.

In other words, zZ means that z is a function z:AαZα such that z(α)Zα for each α. Thus, zX and we have

ZX

as sets.

Theorem 1.

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

Proof.

Let us denote by τX and τZ the product topologies for X and Z, respectively. Also, let

πX,α:XXα,πZ,α:ZZα

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

βX = {πX,α-1(U):αA,Uτ(Xα)},
βZ = {πZ,α-1(U):αA,Uτ(Zα)},

where τ(Xα) is the topology of Xα and τ(Zα) is the subspace topology of ZαXα. The claim follows as

βZ={BZ:BβX}.

Title product topology and subspace topology
Canonical name ProductTopologyAndSubspaceTopology
Date of creation 2013-03-22 15:35:33
Last modified on 2013-03-22 15:35:33
Owner matte (1858)
Last modified by matte (1858)
Numerical id 6
Author matte (1858)
Entry type Theorem
Classification msc 54B10