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. 
 A: 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 $X_1$. 
