T1 is hereditary
This article gives the statement, and possibly proof, of a topological space property (i.e., T1 space) satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces)
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Further information: T1 space
A topological space is termed if it satisfies the following equivalent conditions:
- For any two distinct points , there exists an open subset of such that
- For every point , the singleton subset is closed in
- For every point , the intersection of all open subsets of containing , is
Further information: Subspace topology
The subspace topology on a subset of is defined in the following equivalent ways:
- A subset of is open in iff there exists an open subset of such that .
- A subset of is closed in iff there exists a closed subset of such that .
Proof in terms of two-point definition of T1
Given: A -space , a subset
To prove: is a -space when endowed with the subspace topology
Proof: We need to show that if are both points of , then there exists an open subset of containing and not containing .
Since are distinct points of , they are also distinct points of . Since is a -space, there exists an open subset of such that and . Now consider the set . Then, by definition of subspace topology, is open in . Further, since and , we have . Since , we have . Thus, is an open subset of containing and not containing .