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.