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We prove that a Hausdorff space $X$ is locally compact if and only if its topology coincides with the weak topology induced by $C_{\infty }\left(X\right)$. It is shown that for a Hausdorff space $X$, there exists a locally compact Hausdorff space $Y$ such that $C_{\infty }\left(X\right)\cong C_{\infty }\left(Y\right)$. It is also shown that for locally compact spaces $X$ and $Y$, $C_{\infty }\left(X\right)\cong C_{\infty }\left(Y\right)$ if and only if $X\cong Y$. Prime ideals in $C_{\infty }\left(X\right)$ are uniquely represented by a class of prime ideals in $C^{*}\left(X\right)$. $\infty$-compact spaces are introduced and it turns out that a locally compact space $X$ is $\infty$-compact if and only if every...