False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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FALSE: every Hausdorff compactification has the Stone–Čech extension property

Statement

Every Hausdorff compactification has the Stone–Čech extension property.

Facts & Assumptions

Given: The discrete naturals, their one-point compactification $\mathbb N^*$ , and the parity map $p:\mathbb N\to[0,1]$ .

[L1]

The one-point compactification of a locally compact Hausdorff noncompact space is a compact Hausdorff space in which the original space is dense ( $X^{*}$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{*}$ exactly when $X$ is not compact; and $X^{*}$ is Hausdorff exactly when $X$ is locally compact and Hausdorff).

Refutation

technique · contradiction

1.1

Discrete $\mathbb N$ is locally compact, Hausdorff, and noncompact, so [L1] makes $\mathbb N^*$ a Hausdorff compactification.

1.2

Suppose it had the Stone–Čech property. Its parity map would then extend continuously to $\mathbb N^*$ .

assume-contra

2.1

The same cofinite-neighbourhood argument as follows directly from The one-point (Alexandroff) compactification $X^{*} = X \cup \{\infty\}$ , whose open sets are the open sets of $X$ together with the complements in $X^{*}$ of the closed compact subsets of $X$ : continuity at $\infty$ would make $p$ eventually lie in an interval of radius $1/3$ , while arbitrarily large even and odd naturals have values $0$ and $1$ . This is impossible.

step 1.2

3.1

Therefore this Hausdorff compactification is not Stone–Čech, refuting the displayed statement.

step 1.1step 2.1discharge-contradiction∎

Depends on

Used by

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Sources

E. Moorhouse, The Stone–Čech Compactification (standard reference, not scraped)