Locally normal not implies normal
From Topospaces
This article gives the statement and possibly, proof, of a nonimplication 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 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