Normal Hausdorff space: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[topological space]] is said to be '''normal''' if | 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 subset]]s 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 collection|point-finite]] open cover possesses a [[shrinking]] | |||
===Definition with symbols=== | ===Definition with symbols=== | ||
Revision as of 10:01, 18 August 2007
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 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
Fill this in later
Relation with other properties
Stronger properties
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 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.