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2008-05-11T19:57:51Z

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Vipul at 22:30, 15 December 2007

2007-12-15T22:30:32Z

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{{topospace property non-implication}}

==Statement==

A [[regular space]] need not be [[normal space|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. {{proofat|[[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]]. {{proofat|[[Normality is not product-closed]]}}

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

Regular not implies normal - Revision history

Vipul: 1 revision

Vipul at 22:30, 15 December 2007