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Keywords:
pasting topological spaces at one point; rings of continuous (bounded) real functions on $X$; $z$-ideal; $z^\circ $-ideal; $C$-embedded; $P$-space; $F$-space.
pasting topological spaces at one point; rings of continuous (bounded) real functions on $X$; $z$-ideal; $z^\circ $-ideal; $C$-embedded; $P$-space; $F$-space.
Summary:
Let $\lbrace X_\alpha \rbrace _{\alpha \in \Lambda }$ 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 $C(X)$ according to the same ideals of $C(X_\alpha )$’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 any set of prime ideals of cardinality $m$ in a ring $R$ such that the intersection of these prime ideals can not be obtained as an intersection of fewer than $m$ prime ideals in $R$? Finally, we answer an open question in [11].
References:
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