Regular not implies normal

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.
 * 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.

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.