Pasting topological spaces at one point

Aliabad, Ali Rezaei

Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1193-1206.

Résumé

Let $\{X_{\alpha }{\}}_{\alpha \in \mathrm{\Lambda }}$ be a family of topological spaces and $x_{\alpha }\in X_{\alpha }$, for every $\alpha \in \mathrm{\Lambda }$. 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].

Détail

Classification : 54B15, 54C40, 54C45, 54G05, 54G10
Mots-clés : 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.