Equivalence of Definitions of Irreducible Space

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

1 implies 2
Let $T$ be not the union of any two proper closed subsets.

Suppose $T$ admits a finite cover by proper closed subsets.

By the Well-Ordering Principle, there exists a minimal natural number $n\in\N$ such that $T$ has a cover by $n$ proper closed subsets, say $F_1, \ldots, F_n$.

By definition of proper subset, we must have $n>1$.

By Union of Closed Sets is Closed, $F_{n-1}\cup F_n$ is closed.

By Union is Associative, $S = F_1 \cup \cdots \cup (F_{n-1}\cup F_n)$.

By minimality of $n$, $F_{n-1}\cup F_n$ is not a proper subset.

Thus $F_{n-1}\cup F_n = S$.

This is a contradiction.

Thus $T$ has no finite cover by proper closed subsets.

2 implies 1
This is immediate, because a cover by two sets is a finite cover.