# Hausdorffness is closure-local

From Topospaces

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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Get more facts about Hausdorff space |Get facts that use property satisfaction of Hausdorff space | Get facts that use property satisfaction of Hausdorff space|Get more facts about closure-local property of topological spaces

## Contents

## Statement

Suppose is a topological space such that for any point , there exists an open subset such that the closure is a Hausdorff space. Then, is also a Hausdorff space.

## Related facts

## Facts used

## Proof

**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 .

**Proof**:

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 |