Irreducible Space is Pseudocompact

Theorem

Let $T = \struct {S, \tau}$ be a topological space which is irreducible.


Then $T$ is pseudocompact.


Proof

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

A constant mapping is trivially bounded.

Hence the result by definition of pseudocompact.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness
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