Irreducible and Hausdorff implies one-point space
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.