Regularity is hereditary
This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
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This article gives the statement, and possibly proof, of a basic fact in topology.
Statement
Property-theoretic statement
The property of topological spaces of being a regular space is a hereditary property of topological spaces.
Verbal statement
Any subset of a regular space is regular under the subspace topology.
Definitions used
Regular space
Further information: Regular space
Subspace topology
Further information: Subspace topology
Proof
Proof outline
Any subspace of a T1 space is T1, so we only need to check separation of points and closed sets. We do this as follows:
- Pick a point, and a closed set not containing it, in the subspace (the set is closed relative to the subspace)
- By the definition of subspace topology, find a closed set in the whole space, whose intersection with the subspace is the given closed set)
- Separate the point and this bigger closed set, in the whole space, by disjoint open sets (using regularity of the whole space)
- Intersect these open sets with the subspace to get a separation by disjoint open sets in the subspace
Note that this proof does not work for normality because we would need to enlarge both closed subsets, and the process of enlarging might lead to intersection.