T1 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: T1

This article is about a basic definition in topology.
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Definition

A topological space is termed a $T_{1}$ -space if it satisfies the following equivalent conditions:

Given an ordered pair of distinct points, there is an open subset of the topological space containing the first point but not the second
Every singleton subset is a closed subset (more loosely, all points are closed)

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
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Any subset of a $T_{1}$ -space, is a $T_{1}$ -space under the subspace topology.

Products

This property of topological spaces is closed under taking arbitrary products
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A direct product of $T_{1}$ -spaces is $T_{1}$ .

Refining

This property of topological spaces is preserved under refining, viz, if a set with a given topology has the property, the same set with a finer topology also has the property
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If we take a $T_{1}$ -space, and switch to a [finer topology]], the new space is also $T_{1}$ . This is because the addition of more open sets does not disturb the fact that points are closed.

Local nature

This property of topological spaces is local, in the sense that the topological space satisfies the property if and only if every point has an open neighbourhood which satisfies the property