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

Symbol-free definition

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

1. Given an ordered pair of distinct points, there is an open subset of the topological space containing the first point but not the second
2. Every singleton subset is a closed subset (more loosely, all points are closed)
3. Every point equals the intersection of all open subsets of the space containing that point

Definition with symbols

A topological space $X$ is termed a $T_{1}$-space (or Frechet space or accessible space) if it satisfies the following equivalent conditions:

1. Given two distinct points $x,y\in X$, there exists an open subset $U$ of $X$ such that $x\in U$ and $y\notin U$
2. For every $x\in X$, the singleton set $\{x\}$ is a closed subset
3. For every $x\in X$, the intersection of all open subsets of $X$ containing $\{x\}$ is is precisely $\{x\}$

Examples

Extreme examples

• The empty space is $T_{1}$, because the condition for two points is vacuously satisfied
• The one-point space is also $T_{1}$, because the condition for two points is vacuously satisfied
• More generally, any discrete space -- a topological space where all subsets are open, is $T_{1}$

Examples from metric spaces

• Euclidean space is $T_{1}$: given any two points in Euclidean space, we can make an open set containing the first and not containing the second. Moreover, any subspace of Euclidean space is $T_{1}$
• Any metrizable space is $T_{1}$: In a metric space, we can always take an open ball containing one point and not the other.

Other examples

• Any T0 topological group is $T_{1}$

Relation with other properties

Stronger properties

• Hausdorff space
• US-space

Weaker properties

• Symmetric space
• T0 space

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.
View other subspace-hereditary properties of topological spaces

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
View all properties of topological spaces closed under products

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
View all refining-preserved properties of topological spaces OR View all coarsening-preserved properties of topological spaces

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

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 99 (definition in paragraph)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 26 (formal definition, as part of a list of definitions of separation axioms)