# T1 Space

A topological space is said to be a space if for any pair of distinct points of , there exist two open sets which contain one but not the other.

In other words, a topological space is said to be a space if for any there exist open sets and such that and .

**Example:**

If with topology defined on is not a space because for , we have open sets and such that . This shows that we cannot find an open set which contains but not, so is not a space. But we have already showed that is a space. This shows that a space may not be a space, but the converse is always true.

**Example: **

The real line with usual topology is a space.

To prove this, we suppose that . Further assume that . Since the usual topology on consists of open intervals, we have open sets and , such that and . This shows that the real line with the usual topology is a space.

**Theorems:**

• Every space is a space

• An indiscrete topological space with at least two points is not a space.

• The discrete topological space with at least two points is a space.

• Every two point co-finite topological space is a space.

• Every two point co-countable topological space is a space.

• Every subspace of space is a space.

• A topological space is a space if and only if each of its finite subsets is a closed set.

• The following statements about a topological space are equivalent: (1) is a space; (2) each singleton subset of is closed; (3) each subset A of is the intersection of its open supersets.

• Any homeomorphic image of a space is a space.

• If x is a limit point of a set A in a space , then every open set containing an infinite number of distinct points of .

• A finite set has no limit points in a space.