# Regular 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
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## Statement

A regular space need not be normal. In other words, the property of being regular is strictly weaker than the property of being normal.

## More general statements

There are many ways of seeing that regularity does not imply normality. One is the metaproperty route; we show that there are metaproperties that regularity satisfies which normality doesn't. In particular:

• Regularity is hereditary, but normality isn't. Thus, any example of a subspace of a normal space that is not normal, gives an example of a regular space which is not normal. For full proof, refer: Normality is not hereditary
• Regularity is closed under taking products, but normality isn't. Thus, any example of a product of normal spaces which is not normal, gives an example of a regular space which is not normal. A specific example is the Sorgenfrey plane. For full proof, refer: Normality is not product-closed

Note that these examples also show that completely regular spaces need not be normal, because the property of being completely regular is also closed under taking subspaces and products.