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Let ${\left\{X_{\alpha }\right\}}_{\alpha \in \Lambda }$ be a family of topological spaces and $x_{\alpha }\in X_{\alpha }$, for every $\alpha \in \Lambda$. Suppose $X$ is the quotient space of the disjoint union of $X_{\alpha }$’s by identifying $x_{\alpha }$’s as one point $\sigma$. We try to characterize ideals of $C\left(X\right)$ according to the same ideals of $C\left(X_{\alpha }\right)$’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let $m$ be an infinite cardinal. (1) Is there any ring $R$ and $I$ an ideal in $R$ such that $I$ is an irreducible intersection of $m$ prime ideals? (2) Is there...

Let $I$ be a semi-prime ideal. Then $P_{\circ }\in Min\left(I\right)$ is called irredundant with respect to $I$ if $I\ne {\bigcap }_{P_{\circ }\ne P\in Min\left(I\right)}P$. If $I$ is the intersection of all irredundant ideals with respect to $I$, it is called a fixed-place ideal. If there are no irredundant ideals with respect to $I$, it is called an anti fixed-place ideal. We show that each semi-prime ideal has a unique representation as an intersection of a fixed-place ideal and an anti fixed-place ideal. We say the point $p\in \beta X$ is a fixed-place point if $O^p\left(X\right)$ is a fixed-place ideal. In this situation...

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...