Irreducible Hausdorff Topological Spaces

Question

I was studying Zariski Topology, and we know that a Zariski topological space is not Hausdorff, now I want to know what are the irreducible Hausdorff topological spaeces?

Any comments would be highly appreciated.

Answer 1

If $X$ has two or more points (Say $p \neq q$ are both in $X$) and is Hausdorff, then it there are open sets $U,V$ of $x$ with $p \in U$, $q \in V$ and $ U \cap V = \emptyset$.

But then $X = (X\setminus U) \cup (X \setminus V)$ (follows from $U \cap V = \emptyset$) and both are proper closed subsets of $X$ ($p$ and $q$ are in the respective complements). So $X$ is not irreducible.

So for Hausdorff spaces we have $|X| \ge 2 \Rightarrow$ $X$ is not irreducible. Or equivalently: $X$ irreducible $\Rightarrow |X| \le 1$.

So the only non-empty Hausdorff irreducible space is a singleton (in its unique topology).