T1 space

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 $$\{ x \}$$

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 $$T_1$$

Stronger properties

 * Hausdorff space
 * US-space

Weaker properties

 * Symmetric space
 * T0 space

Metaproperties
Any subset of a $$T_1$$-space, is a $$T_1$$-space under the subspace topology.

A direct product of $$T_1$$-spaces is $$T_1$$.

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.

Textbook references

 * , Page 99 (definition in paragraph)
 * , Page 26 (formal definition, as part of a list of definitions of separation axioms)