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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