Locally normal not implies normal

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
View a complete list of topological space property non-implications | View a complete list of topological space property implications |Get help on looking up topological space property implications/non-implications
|

Statement

A locally normal space need not be normal. In fact, it need not be normal even if it is assumed to be completely regular.

Example

• A trivial example of a locally normal space that is not normal is the line with two origins -- this is not even Hausdorff
• An example of a locally normal space that is completely regular and yet not normal is the Moore plane