# 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.