# T1 Space

A topological space $$X$$ is said to be a $${T_1}$$ space if for any pair of distinct points of $$X$$, there exist two open sets which contain one but not the other.

In other words, a topological space $$X$$ is said to be a $${T_1}$$ space if for any $$x,y \in X,{\text{ }}x \ne y$$ there exist open sets $$U$$ and $$V$$ such that $$x \in U,{\text{ }}y \notin U$$ and $$y \in V,{\text{ }}x \notin V$$.

**Example:**

If $$X = \left\{ {a,b,c} \right\}$$ with topology $$\tau = \left\{ {\phi ,X,\left\{ a \right\},\left\{ b \right\},\left\{ {a,b} \right\}} \right\}$$ defined on $$X$$ is not a $${T_1}$$ space because for $$a,c \in X$$, we have open sets $$\left\{ a \right\}$$ and $$X$$ such that $$a \in \left\{ a \right\},{\text{ }}c \notin \left\{ a \right\}$$. This shows that we cannot find an open set which contains $$c$$ but not$$a$$, so $$\left( {X,\tau } \right)$$ is not a $${T_1}$$ space. But we have already showed that $$\left( {X,\tau } \right)$$ is a $${T_o}$$ space. This shows that a $${T_o}$$ space may not be a $${T_1}$$ space, but the converse is always true.

**Example: **

The real line $$\mathbb{R}$$ with usual topology is a $${T_1}$$ space.

To prove this, we suppose that $$x,y \in \mathbb{R},{\text{ }}x \ne y$$. Further assume that $$x < y$$. Since the usual topology on $$\mathbb{R}$$ consists of open intervals, we have open sets $$U = \left] { – \infty ,{\text{ }}y} \right[$$ and $$V = \left] {x,{\text{ }}\infty } \right[$$, such that $$x \in U,{\text{ }}y \notin U$$ and $$y \in V,{\text{ }}x \notin V$$. This shows that the real line $$\mathbb{R}$$ with the usual topology is a $${T_1}$$ space.

**Theorems:**

• Every $${T_1}$$ space is a $${T_o}$$ space

• An indiscrete topological space with at least two points is not a $${T_1}$$ space.

• The discrete topological space with at least two points is a $${T_1}$$ space.

• Every two point co-finite topological space is a $${T_1}$$ space.

• Every two point co-countable topological space is a $${T_1}$$ space.

• Every subspace of $${T_1}$$ space is a $${T_1}$$ space.

• A topological space is a $${T_1}$$ space if and only if each of its finite subsets is a closed set.

• The following statements about a topological space $$X$$ are equivalent: (1) $$X$$ is a $${T_1}$$ space; (2) each singleton subset of $$X$$ is closed; (3) each subset A of $$X$$ is the intersection of its open supersets.

• Any homeomorphic image of a $${T_1}$$ space is a $${T_1}$$ space.

• If x is a limit point of a set A in a $${T_1}$$ space $$X$$, then every open set containing an infinite number of distinct points of $$A$$.

• A finite set has no limit points in a $${T_1}$$ space.

Zia fatima

July 7@ 10:50 amEvery T1 subspace is T1 space