Pasting topological spaces at one point. (English) Zbl 1164.54338

Summary: Let \(\{X_\alpha \}_{\alpha \in \Lambda }\) 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 the \(X_\alpha \)’s by identifying the \(x_\alpha \)’s as one point \(\sigma \). We try to characterize ideals of \(C(X)\) according to the same ideals of the \(C(X_\alpha )\)’s. In addition we generalize the concept of rank of a point 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 \(R\)?


54C40 Algebraic properties of function spaces in general topology
54C45 \(C\)- and \(C^*\)-embedding
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
54G10 \(P\)-spaces
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