Irreducible Space is Pseudocompact

Theorem
Let $T = \left({X, \vartheta}\right)$ be a topological space which is hyperconnected.

Then $T$ is pseudocompact.

Proof
We have that Continuous Real-Valued Function on Hyperconnected Space is Constant.

A constant mapping is trivially bounded.

Hence the result by definition of pseudocompact.