# To Space

A topological space is said to be a space if for any pair of distinct points of , there exists at least one open set which contains one of them but not the other.

In other words, a topological space is said to be a space if for any , there exists an open set such that but .

**Example:**

Let with topology defined on , then is a space, because:

**(1)** for and , there exists an open set such that and

**(2)** for and , there exists an open set such that and

**(3)** for and , there exists an open set such that and .

**Theorems**

• Let be an indiscrete topological space with at least two points, then is not a space.

• Let be a discrete topological space with at least two points, then is not a space.

• The real line with usual topology is a space.

• Every sub space of space is a space.

• A topological space is a space if and only if for any .

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

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

• If each singleton subset of a two point topological space is closed, then it is a space.

• If each finite subset of a two point topological space is closed, then it is a space.

• Any homeomorphic image of a space is a space.

• A pseudo metric space is a metric space if and only if it is a space.