characterization of subspace topology


Let X be a topological spaceMathworldPlanetmath and YX any subset. The subspace topology on Y is the weakest topology making the inclusion mapMathworldPlanetmath continuousMathworldPlanetmathPlanetmath.


Let 𝒮 denote the subspace topology on Y and j:YX denote the inclusion map.

Suppose {𝒯ααJ} is a family of topologies on Y such that each inclusion map jα:(Y,𝒯α)X is continuous. Let 𝒯 be the intersectionMathworldPlanetmath αJ𝒯α. Observe that 𝒯 is also a topology on Y. Let U be open in X. By continuity of jα, the set jα-1(U)=j-1(U) is open in each 𝒯α; consequently, j-1(U) is also in 𝒯. This shows that there is a weakest topology on Y making inclusion continuous.

We claim that any topology strictly weaker than 𝒮 fails to make the inclusion map continuous. To see this, suppose 𝒮0𝒮 is a topology on Y. Let V be a set open in 𝒮 but not in 𝒮0. By the definition of subspace topology, V=UY for some open set U in X. But j-1(U)=V, which was specifically chosen not to be in 𝒮0. Hence 𝒮0 does not make the inclusion map continuous. This completesPlanetmathPlanetmathPlanetmathPlanetmath the proof. ∎

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