## 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 $$R$$?

### MSC:

 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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### References:

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