Regular not implies normal

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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.