To Space

A topological space X is said to be a {T_o} space if for any pair of distinct points of X, there exist 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 exist 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 exist 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 exist 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 exist an open set \left\{ b \right\} such that b \in \left\{ b \right\} and c \notin \left\{ b \right\}.

Theorems:
• Let X is an indiscrete topological space with at least two points, then X is not a {T_o} space.
• Let X is 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 {T_o} space.
• Every sub space of {T_o} space is {T_o} space.
• A topological space X is {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 {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.

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