characterization of subspace topology
Theorem.
Let be a topological space and any subset. The subspace topology on is the weakest topology making the inclusion map
continuous
.
Proof.
Let denote the subspace topology on and denote the inclusion map.
Suppose is a family of topologies on such that each inclusion map is continuous. Let be the intersection . Observe that is also a topology on . Let be open in . By continuity of , the set is open in each ; consequently, is also in . This shows that there is a weakest topology on making inclusion continuous.
We claim that any topology strictly weaker than fails to make the inclusion map continuous. To see this, suppose is a topology on . Let be a set open in but not in . By the definition of subspace topology, for some open set in . But , which was specifically chosen not to be in . Hence does not make the inclusion map continuous. This completes the proof.
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Title | characterization of subspace topology |
---|---|
Canonical name | CharacterizationOfSubspaceTopology |
Date of creation | 2013-03-22 15:40:32 |
Last modified on | 2013-03-22 15:40:32 |
Owner | mps (409) |
Last modified by | mps (409) |
Numerical id | 6 |
Author | mps (409) |
Entry type | Theorem |
Classification | msc 54B05 |