show/hide this revision's text 2 strikethrough wrong example

I might have a counterexample, inspired by Greg Kuperberg's reference to Eric Wofsey's answer to that other question.

If $Y$ is a one point space, the question becomes whether every homomorphism from $C(X)$ to $\mathbb{C}$ is evaluation at a point. By Eric Wofsey's aforementioned answer, to search for a counterexample to this is to search for a hemicompact k-space $X$ such that there exists a point in $\beta X$ to which every element of $C(X)$ extends continuously. He gives an example of a space satisfying the latter property: $\omega_1$, the first uncountable ordinal. A desk reference tells me that $\omega_1$ is locally compact and Lindelof; hence it has a cover by a sequence of open sets with compact closure, implying it is a hemicompact k-space.

Edit: Nope, my ignorance of $\omega_1$ was showing. Corrected promptly by Greg Kuperberg.
show/hide this revision's text 1

I might have a counterexample, inspired by Greg Kuperberg's reference to Eric Wofsey's answer to that other question.

If $Y$ is a one point space, the question becomes whether every homomorphism from $C(X)$ to $\mathbb{C}$ is evaluation at a point. By Eric Wofsey's aforementioned answer, to search for a counterexample to this is to search for a hemicompact k-space $X$ such that there exists a point in $\beta X$ to which every element of $C(X)$ extends continuously. He gives an example of a space satisfying the latter property: $\omega_1$, the first uncountable ordinal. A desk reference tells me that $\omega_1$ is locally compact and Lindelof; hence it has a cover by a sequence of open sets with compact closure, implying it is a hemicompact k-space.