product topology and subspace topology
Let with be a collection of topological spaces, and let be subsets. Let
and
In other words, means that is a function such that for each . Thus, and we have
as sets.
Theorem 1.
The product topology of coincides with the subspace topology induced by .
Proof.
Let us denote by and the product topologies for and , respectively. Also, let
be the canonical projections defined for and . The subbases (http://planetmath.org/Subbasis) for and are given by
where is the topology of and is the subspace topology of . The claim follows as
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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 |