Irreducible topological spaces and irreducible Hausdorff space A topological space $X$ is said to be irreducible if it cannot be written as a union $X = A∪B$, where $A$ and $B$ are proper closed subspaces of $X$
I want to try the following:
Show:
$(i)$ That every set with the indiscrete topology is irreducible, and every infinite set with the finite complement topology is irreducible.
$(ii)$ That an irreducible Hausdorff space contains at most one point.
Can you give me any suggestions on how to try this?
 A: HINT: Prove that a space $X$ is irreducible if and only it does not contain two disjoint non-empty open sets.
A: An irreducible topological space $ X $ can be characterized as:
(a) $X$ is irreducible if and only if every pair of nonempty open subsets has a nonempty intersection.
$(i)$ For the first part, it suffices to note that a space with an indiscrete topology has no nonempty proper closed subspaces. For the second part, note that if $X$ is an infinite set with the finite complement topology, then the closed proper subsets are precisely the finite subsets of $ X$, and the union of two such subsets is always finite and this is always a proper subset of  $X$. Therefore $ X$ cannot be written as the union of two closed proper subsets.
$(ii)$ If $X$ is a Hausdorff space and $u, v \in X$ then one can find open subsets $U$ and $V$ such that $u \in U$ and $v \in V$ (hence both are nonempty) such that $U ∩ V = \emptyset$. Therefore $X$ is not irreducible because it does not satisfy the characterization of such spaces in the first part of $(a)$  above.
