brief explanation of any compactification of a locally compact Hausdorff space X is a quotient space of its Stone-Chech Compactification

Question

I want a simple explanation(not rigors proof; maybe for the rigors proof references are fine) of any compactification of a locally compact Hausdorff space $X$ is a quotient space of its Stone-Chech Compactification.

What I know is every normal space has a compactification[but not fully understand its proof], and if $X$ is locally compact Hausdorff there exists one-point compactification $Y=X \cup \{\infty\}$ unique up to homeomorphism. [one point compactifications are fine! I understand the procedures of one-point compactification from Croom's topology textbook.]

Since my starting point is a compactification of a locally compact Hausdorff space, so I can simply consider one-point compactification of $X$, and then prove or find a plausible reason for this space $Y=X\cup\{\infty\}$ is a quotient space of its Stone-Chech Compactification.

I just know the definition of Stone-Cech compactification from Wikipedia but do not fully understand its definition yet.

Answer 1

If $K$ is any compactification of $X$ (by which I mean a compact Hausdorff space with $j:X\to K$ a dense embedding), then the definition of the Stone-Cech-compactification provides you with a continuous map $J:\beta X\to K$ such that $J\circ i = j$, where $i:X\to\beta X$ is the embedding of $X$ into its Stone-Cech-compactification. This is the crucial aspect to know about the Stone-Cech-compactification: Any other map from X into a compact Hausdorff space factors through $\beta X$. Now $J$ is a continuous map between two compact Hausdorff spaces. To see that it is a quotient map you just need to prove the following:

1. $J$ is surjective (this follows as $\beta X$ is compact, $j(X)$ is dense in $K$ and $J(\beta X)$ consequently is dense in $K$ as well)
2. If $J^{-1}(U)$ is open for $U\subseteq K$, then $U$ is open. This is true for any surjective map between compact Hausdorff spaces and follows from the fact that images of compact sets are compact.

I don't quite understand your remark where you want to consider the one-point-compactification only. This is just a special case that generalises as sketched above (usually there are many different compactifications of a space, not just the two mentioned so far), but is not sufficient in itself.