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:

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 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 nota, 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 {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, so 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 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 {T_1} space.
• A topological space is a {T_1} space if and only if its each finite subsets is a closed set.
• Following statements about a topological space X are equivalent. (1) X ia 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.

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