# Irreducible and Hausdorff implies one-point space

From Topospaces

## Statement

Suppose is a non-empty topological space that is both an irreducible space and a Hausdorff space. Then, must be a one-point space.

## Related facts

### Similar facts

## Proof

We prove a somewhat modified form: we prove that an Hausdorff space containing two distinct points cannot be irreducible.

**Given**: A non-empty topological space that is Hausdorff and has at least two points. Say are two distinct points of .

**To prove**: is not irreducible, i.e., it contains two non-empty open subsets with empty intersection.

**Proof**: Use the definition of Hausdorff to find non-empty open subsets that have empty intersection. This completes the proof.