Pasting topological spaces at one point

Ali Rezaei Aliabad

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 4, page 1193-1206
  • ISSN: 0011-4642

Abstract

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Let be a family of topological spaces and , for every . Suppose is the quotient space of the disjoint union of ’s by identifying ’s as one point . We try to characterize ideals of according to the same ideals of ’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let be an infinite cardinal. (1) Is there any ring and an ideal in such that is an irreducible intersection of prime ideals? (2) Is there any set of prime ideals of cardinality in a ring such that the intersection of these prime ideals can not be obtained as an intersection of fewer than prime ideals in ? Finally, we answer an open question in [11].

How to cite

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Aliabad, Ali Rezaei. "Pasting topological spaces at one point." Czechoslovak Mathematical Journal 56.4 (2006): 1193-1206. <http://eudml.org/doc/31099>.

@article{Aliabad2006,
abstract = {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].},
author = {Aliabad, Ali Rezaei},
journal = {Czechoslovak Mathematical Journal},
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; rings of continuous (bounded) real functions on  ; -ideal},
language = {eng},
number = {4},
pages = {1193-1206},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Pasting topological spaces at one point},
url = {http://eudml.org/doc/31099},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Aliabad, Ali Rezaei
TI - Pasting topological spaces at one point
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 4
SP - 1193
EP - 1206
AB - 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].
LA - eng
KW - 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; rings of continuous (bounded) real functions on  ; -ideal
UR - http://eudml.org/doc/31099
ER -

References

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