Why can any homeomorphism on a topological space can be extended to its Čech-Stone compactification?

Question

Let $X$ be a topological space.

More times, I came across the statement that the Čech-Stone compactification (the "most general" compactification of a topological space) clearly has a property that any homeomorphism on the original space can be extended onto the compactification. Why is that clear though? How do we check that homeomorphisms can be extended?

My idea would be:

1. Take homeomorphism on the original space
2. Define the homeomorphism on the remainder (what was "added" to the space to create the $\beta X$ compactification)

Am I right? Could you provide some examples of particular homeomorphisms and how to construct the extension? (I don´t mean the simplest cases as adding just one infinity point to the original space, but something more interesting, ideally for the Čech-Stone compactification).

Answer 1

Recall the universal mapping property for $\beta X$: we have an embedding $e: X \to \beta X$ and for every continuous map $f$ from $X$ to $Y$ where $Y$ is any compact Hausdorff space there is a unique map $\beta f: \beta X\to Y$ which satisfies $\beta f \circ e = f$. If we consider $e$ to be the identity for convenience sake, that just says that $\beta f$ extends $f$ as a map.

If now $h: X \to X$ is a homeomorphism with inverse $h’: X \to X$, then we can consider $h$ to be a map from $X$ to $\beta X$ as well (if we assume $X \subseteq \beta X$, and it’s still continuous, of course. (Being formal, we really mean $e \circ h: X \to \beta X$).

We apply the universal property (as $\beta X$ is compact Hausdorff, we can) and get $\beta h: \beta X \to \beta X$ extending $h$.

Similarly we get $\beta h’:\beta X \to \beta X$ and as $h’ \circ h = 1_X$ and so $\beta h’ \circ \beta h$ extends $1_X$ and by unicity (and the fact that $X$ is dense in $\beta X$) we can conclude that $\beta h’ \circ \beta h = 1_{\beta X}$ and so $\beta h$ is a homeomorphism of $\beta X$ extending $h$.

From an abstract point of view: $X \mapsto \beta X$ is a functor from the category of Tychonoff spaces to the category of compact Hausdorff spaces and as such it preserves isomorphisms.