Normal Hausdorff space

''There are two alternative definitions of the term. Please see:'' Convention:Hausdorffness assumption

Equivalent definitions in tabular format
Note that in each of the definitions, the T1 space assumption (that points are closed) can be replaced by the (a priori stronger) Hausdorff space assumption, without changing the meaning of the overall definition.

Some people use the term normal space for what is called here a normal Hausdorff space; however, we define the term normal space as not having the T1 space assumption.

Equivalence of definitions
The direction (2) implies (1) is easy: if there is a continuous function $$f:X \to [0,1]$$ such that $$A \subseteq f^{-1}(\{ 0 \})$$ and $$B \subseteq f^{-1}(\{ 1 \})$$, then we can take the open sets $$f^{-1}((0,1/2))$$ and $$f^{-1}((1/2,1))$$.

The direction (1) implies (2) follows from Urysohn's lemma.

Metaproperties
Below is more information:

A product of two normal spaces, endowed with the product topology, need not be normal.

It is possible to have a normal space $$X$$ and a subspace $$Y$$ of $$X$$ such that $$Y$$ is not a normal space.

Any subspace of a normal space need not be normal. However, any closed subset of a normal space is normal, under the subspace topology.

Moving to a finer topology, i.e., adding more open subsets, may destroy the property of normality.

Facts

 * Any connected normal space having at least two points (and more generally, any connected Urysohn space having at least two points) is uncountable.

Effect of property operators
A topological space can be realized as a subspace of a normal space iff it is completely regular. Necessity follows from the fact that normal spaces are completely regular, and any subspace of a completely regular space is completely regular. Sufficiency follows from the Stone-Cech compactification.

A topological space in which every subspace is normal is termed hereditarily normal (some people call it completely normal). Note that metrizable spaces are hereditarily normal.

Textbook references

 * , Page 195,Chapter 4, Section 31 (formal definition, along with definition of regular space)
 * , Page 28 (formal definition)