Hausdorffness is closure-local

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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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Statement

Suppose X is a topological space such that for any point x \in X, there exists an open subset U \ni x such that the closure \overline{U} is a Hausdorff space. Then, X is also a Hausdorff space.

Related facts

Facts used

  1. Openness is transitive

Proof

Given: A topological space X such that for any point x \in X, there exists an open subset U \ni x such that the closure \overline{U} is a Hausdorff space. We are given two distinct points x,y \in X.

To prove: We can find disjoint open subsets V,W of X such that V \ni x, W \ni y.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 There exists an open subset U \ni x such that the closure \overline{U} is a Hausdorff space. given data directly used direct
2 If y \notin \overline{U}, then we can take V = U and W = X \setminus \overline{U}. Step (1)
3 If y \in \overline{U}, there exist open subsets A,B of \overline{U} such that x \in A, y \in B and A \cap B is empty.
4 Continuing with the assumption and notation from Step (3), then there exists W open in X such that B = W \cap \overline{U}. In particular, y \in W. definition of subspace topology Step (3)
5 Continuing with the assumption and notation from Step (3), the set V = A \cap U is open in X and contains x. Fact (1) Steps (1), (3) By definition of subspace topology going from \overline{U} to U, V is open in U. Thus, by Fact (1), V is open in X. Also, x \in V because x \in A (Step (3)) and x \in U (Step (1)).
6 Continuing with the assumption and notation from Steps (3)-(5), the sets V and W are disjoint. Steps (3), (4) ,(5) V is contained in U by Step (5) and hence in \overline{U}. So, V \cap W = V \cap (W \cap \overline{U}) = V \cap B \subseteq A \cap B. A and B are disjoint by Step (3), hence the intersection is trivial.
7 V,W are the desired disjoint open subsets. Steps (4), (5), (6) Step-combination direct