# T2 Space or Hausdorff Space

A Hausdorff space is a topological space in which each pair of distinct points can be separated by a disjoint open set.

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

**Example:**

Let be a non-empty set with topology (all the subsets of , powers set or discrete topology). Hence:

For

For

For and is a space.

For

For

For .

**Theorems**

• Every metric space is a Hausdorff space.

• Every space is a space but the converse may not be true.

• Every subspace of a space is a space.

• In a Hausdorff space, any point and a disjoint compact subspace can be separated by open sets, in the sense that there exist disjoint open sets, and one contains the point and the other contains the compact subspace.

• Every compact subspace of a Hausdorff space is closed.

• A one-to-one continuous mapping of a compact onto a Hausdorff space is homeomorphism.

• Let , be continuous functions from a space to a Hausdorff space and suppose that for all in a dense subset of . Then for all in .

• A space is a Hausdorff space if and only if every point of is the intersection of its closed neighborhoods.

• Let be a topological space and a Hausdorff space. Let and be a continuous function from to , then the set is a closed set.