Stone–Čech compactification whose input space is more general than being completely regular, and whose output space just locally compact Hausdorff?

Question

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!

Answer 1

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.