Normal Hausdorff space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T4

Definition

Symbol-free definition

A topological space is said to be normal if all points in it are closed sets, and given any two disjoint closed sets in the topological space, there are disjoint open sets containing them.

Definition with symbols

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Relation with other properties

Stronger properties

• Compact Hausdorff space
• Hereditarily normal space
• Perfectly normal space
• Metrizable space
• CW-space

Weaker properties

• Completely regular space
• Regular space
• Hausdorff space
• T1 space
• Kolmogorov space

Metaproperties

Products

NO: This property of topological spaces is not a product-closed property of topological spaces: a product of topological spaces, each satisfying the property, when equipped with the product topology, does not necessarily satisfy the property.
View other properties that are not product-closed

A direct product of normal spaces need not be normal. For full proof, refer: Normality is not direct product-closed

Weak hereditariness

This property of topological spaces is weakly hereditary or closed subspace-closed; in other words, any closed subset (equipped with the subspace topology) of a space with the property, also has the property.
View all weakly hereditary properties of topological spaces | View all subspace-hereditary properties of topological spaces

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