Locally Hausdorff not implies Hausdorff

This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property need not satisfy the second topological space property
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Statement

A locally Hausdorff space need not be Hausdorff.

Example

An example is the line with two origins, which illustrates the more general fact that a locally Euclidean space need not be Hausdorff, and hence need not be a manifold.