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$\sigma $-compact; pseudocompact; $\infty $-compact; $\infty $-compactification; $\operatorname{P}_{\infty }$-space; P-point; regular ring; fixed and free ideals
We prove that a Hausdorff space $X$ is locally compact if and only if its topology coincides with the weak topology induced by $C_\infty (X)$. It is shown that for a Hausdorff space $X$, there exists a locally compact Hausdorff space $Y$ such that $C_\infty(X)\cong C_\infty(Y)$. It is also shown that for locally compact spaces $X$ and $Y$, $C_\infty(X)\cong C_\infty(Y)$ if and only if $X\cong Y$. Prime ideals in $C_\infty(X)$ are uniquely represented by a class of prime ideals in $C^*(X)$. $\infty$-compact spaces are introduced and it turns out that a locally compact space $X$ is $\infty$-compact if and only if every prime ideal in $C_\infty(X)$ is fixed. The existence of the smallest $\infty$-compact space in $\beta X$ containing a given space $X$ is proved. Finally some relations between topological properties of the space $X$ and algebraic properties of the ring $C_\infty(X)$ are investigated. For example we have shown that $C_\infty(X)$ is a regular ring if and only if $X$ is an $\infty$-compact $\operatorname{P}_\infty$-space.
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