Hausdorffness is closure-local
This article gives the statement, and possibly proof, of a topological space property (i.e., Hausdorff space) satisfying a topological space metaproperty (i.e., closure-local property of topological spaces)
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Given: A topological space such that for any point , there exists an open subset such that the closure is a Hausdorff space. We are given two distinct points .
To prove: We can find disjoint open subsets of such that .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||There exists an open subset such that the closure is a Hausdorff space.||given data directly used||direct|
|2||If , then we can take and .||Step (1)|
|3||If , there exist open subsets of such that and is empty.|
|4||Continuing with the assumption and notation from Step (3), then there exists open in such that . In particular, .||definition of subspace topology||Step (3)|
|5||Continuing with the assumption and notation from Step (3), the set is open in and contains .||Fact (1)||Steps (1), (3)||By definition of subspace topology going from to , is open in . Thus, by Fact (1), is open in . Also, because (Step (3)) and (Step (1)).|
|6||Continuing with the assumption and notation from Steps (3)-(5), the sets and are disjoint.||Steps (3), (4) ,(5)||is contained in by Step (5) and hence in . So, . and are disjoint by Step (3), hence the intersection is trivial.|
|7||are the desired disjoint open subsets.||Steps (4), (5), (6)||Step-combination direct|