# To Space

A topological space $$X$$ is said to be a $${T_o}$$ space if for any pair of distinct points of $$X$$, there exists at least one open set which contains one of them but not the other.

In other words, a topological space $$X$$ is said to be a $${T_o}$$ space if for any $$x,y \in X,{\text{ }}x \ne y$$, there exists an open set $$U$$ such that $$x \in U$$ but $$y \notin U$$.

**Example:**

Let $$X = \left\{ {a,b,c} \right\}$$ with topology $$\tau = \left\{ {\phi ,X,\left\{ a \right\},\left\{ b \right\},\left\{ {a,b} \right\}} \right\}$$ defined on $$X$$, then $$\left( {X,\tau } \right)$$ is a $${T_o}$$ space, because:

**(1)** for $$a$$ and $$b$$, there exists an open set $$\left\{ a \right\}$$ such that $$a \in \left\{ a \right\}$$ and $$b \notin \left\{ a \right\}$$

**(2)** for $$a$$ and $$c$$, there exists an open set $$\left\{ a \right\}$$ such that $$a \in \left\{ a \right\}$$ and $$c \notin \left\{ a \right\}$$

**(3)** for $$b$$ and $$c$$, there exists an open set $$\left\{ b \right\}$$ such that $$b \in \left\{ b \right\}$$ and $$c \notin \left\{ b \right\}$$.

**Theorems**

• Let $$X$$ be an indiscrete topological space with at least two points, then $$X$$ is not a $${T_o}$$ space.

• Let $$X$$ be a discrete topological space with at least two points, then $$X$$ is not a $${T_o}$$ space.

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

• Every sub space of $${T_o}$$ space is a $${T_o}$$ space.

• A topological space $$X$$ is a $${T_o}$$ space if and only if for any $$a,b \in X,{\text{ }}a \ne b \Rightarrow \overline {\left\{ a \right\}} \ne \overline {\left\{ b \right\}} $$.

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

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

• If each singleton subset of a two point topological space is closed, then it is a $${T_o}$$ space.

• If each finite subset of a two point topological space is closed, then it is a $${T_o}$$ space.

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

• A pseudo metric space is a metric space if and only if it is a $${T_o}$$ space.