Normality is not product-closed

Statement

A product of normal spaces need not be normal.

Example

An example is the Sorgenfrey plane, which is a product of two copies of the Sorgenfrey line. For full proof, refer: Sorgenfrey line is normal, Sorgenfrey plane is not normal

Consequences

Since the property of being a completely regular space is closed under products, this gives a proof that the property of normality is strictly stronger than the property of complete regularity. Specifically, any example of a product of normal spaces which is not normal, is also an example of a completely regular space which is not normal.

Related facts

Binormality

Closely related to the fact that normality is not product-closed is the notion of a binormal space. A binormal space is a normal space whose product with the unit interval is also normal.

Stronger properties which are closed under products

Although normality is not closed under taking products, there are a number of properties slightly stronger than normality, which are closed under taking products. For instance:

• Compact Hausdorff space
• Metrizable space

Although a product of paracompact Hausdorff spaces need not be paracompact Hausdorff, it is true that a product of a paracompact Hausdorff space with a compact Hausdorff space is paracompact Hausdorff, and hence normal. Thus, paracompact Hausdorff spaces are binormal.