product topology and subspace topology

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




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


as sets.

Theorem 1.

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


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


be the canonical projections defined for X and Z. The subbases ( 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


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