Normal Hausdorff space
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
In the T family (properties of topological spaces related to separation axioms), this is called: T4
This article is about a basic definition in topology.
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Definition
Symbol-free definition
A topological space is said to be normal if it satisfies the following equivalent conditions:
- All points in it are closed sets, and given any two disjoint closed subsets in the topological space, there are disjoint open sets containing them.
- All points in it are closed sets, and given any two disjoint closed subsets, there is a continuous function taking the value 0 at one closed set and 1 at the other
- All points are closed, and every point-finite open cover possesses a shrinking
Definition with symbols
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Relation with other properties
This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied
Stronger properties
- Compact Hausdorff space
- Hereditarily normal space
- Paracompact Hausdorff space
- Regular Lindelof space
- Perfectly normal space
- Metrizable space
- CW-space
- Linearly orderable space
- Collectionwise normal space
Weaker properties
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 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.
Effect of property operators
The hereditarily operator
Applying the hereditarily operator to this property gives: hereditarily normal space
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.