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The Rudin version of Riesz-Markov representation theorem assume that $X$ is locally compact Hausdorff. On the other hand, the input of Stone–Čech compactification theorem is completely regular space $X$, while its output a compact Hausdorff space $\beta X$, i.e.,

Let $X$ be a completely regular space. Then there exists a compact Hausdorff space $\beta X$ and a map $T:X \to \beta X$ such that

  • $T$ is a homeomorphism from $X$ to $T(X)$.

  • $T(X)$ is dense in $\beta X$.

  • If $f: X \to Y$ is continuous with $Y$ being a compact Hausdorff space, then there is a unique continuous map $g: \beta X \to Y$ such that $f= g \circ T$.

Clearly, if a space is locally compact Hausdorff, then it is completely regular. But the Riesz-Markov theorem already applies to locally compact Hausdorff space. So we don't need Stone–Čech compactification theorem.

Is there any version of Stone–Čech compactification theorem such that

  • the input space is more general than complete regularity space, and
  • the output space is locally compact Hausdorff?

If such version of Stone–Čech compactification theorem exists, then we can combine it with Rudin's version of Riesz-Markov representation theorem.

Thank you so much for your help!

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1 Answer 1

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No. If a space admits an embedding into a locally compact Hausdorff space, then it must be completely regular because all subpaces of completely regular spaces are completely regular.

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