Irreducible Space is Pseudocompact
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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