# Which spaces are characterized by functions with compact support ?

It's well known that two locally compact Hausdorff spaces $X, Y$ are homeomorphic iff the rings $C_0(X), C_0(Y)$ (continuous functions vanishing at infinity) are isomorphic.

Is there a class $\mathcal{C}$ of topological spaces such that $X, Y \in \mathcal{C}$ are homeomorphic, iff the rings $C_c(X), C_c(Y)$ are isomorphic ?

Here $C_c(X) = \lbrace f:X \to \mathbb R\mid \operatorname{cl}_X \lbrace x \in X \mid f(x) \neq 0\rbrace\text{ is compact}\;\rbrace$ denotes the ring of continuous functions with compact support.

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Besides containing the compact spaces, what other properties do you want ${\mathcal C}$ to have? (I guess, really, what kind of category are you after?) –  Yemon Choi Mar 22 '12 at 0:32
I have no clue if there is such a $\mathcal{C}$ and if, what properties its spaces have. Assume the question had asked for $C_0$ instead of $C_c$. Then, I think it would be difficult to guess that the property is "locally compact". –  Ralph Mar 22 '12 at 0:51
My point was more that you seem to implicitly assume some kind of maximality or closure property of your class. Otherwise you have silly examples, such as the class of all infinite compact Hausdorff spaces, together with an infinite set with indiscrete topology. What do you actually mean by "class"? –  Yemon Choi Mar 22 '12 at 4:38
Can we say the 'iff' requirement in question will take care of other conditions on the class $\mathcal{C}$? Another question: Does the answer depend on topological considerations on the function spaces also or is it purely algebraic? –  Uday Mar 22 '12 at 7:23
@Yemon: One condition it would be nice to have is that all rings that can be realized are realized in $\mathcal{C}$. I don´t know if this is true for the locally compact vs. $C_0$ case. –  Ramiro de la Vega Mar 22 '12 at 11:55

My naive feeling is that the answer is simply the class of locally compact Hausdorff spaces, for the following reasons. First, for a locally compact Hausdorff space $X$, one can recover $C_0(X)$ from $C_c(X)$ by completion in the uniform norm; and the uniform norm is an algebraic feature because it can be derived from the characters on the ring $C_c(X)$. So for $X$ and $Y$ locally compact Hausdorff, if $C_c(X)$ and $C_c(Y)$ are isomorphic then $X$ and $Y$ are homeomorphic.
Now suppose that $X$ is any completely regular (Hausdorff) space (and surely it is topological spaces in this class that we are interested in). Then $X$ is the disjoint union of an open locally compact subset $X_0$ consisting of points which have a compact neighbourhood and a closed subset $X_1$ consisting of points which do not. Each $f\in C_c(X)$ vanishes on $X_1$ so we are not going to get any information about $X_1$ from $C_c(X)$.
So my feeling is that $C_c(X)$ determines $X_0$ up to isomorphism but gives no information about $X_1$.
To complement this answer: The ring $C_c(X)$ can be directly recovered from $C_0(X)$ as the smallest dense (in the uniform norm) ideal of $C_0(X)$. Say $I$ is a dense ideal of $C_0(X)$ and $f$ is a positive function in $C_0(X)$. Choose $g$ in $I$ such that $|f-g|<\epsilon$. Then one can show that $(f-\epsilon)_+$ is a multiple of $g$ and so belongs to $I$. In this way we get that $C_c(X)$ is contained in $I$. –  Leonel Robert Mar 25 '12 at 21:51