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.