Geodesic
Pasting topological spaces at one point
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1193-1206 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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].
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].
Classification : 54B15, 54C40, 54C45, 54G05, 54G10
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. 
@article{CMJ_2006_56_4_a8,
     author = {Aliabad, Ali Rezaei},
     title = {Pasting topological spaces at one point},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1193--1206},
     year = {2006},
     volume = {56},
     number = {4},
     mrnumber = {2280803},
     zbl = {1164.54338},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a8/}
}
TY  - JOUR
AU  - Aliabad, Ali Rezaei
TI  - Pasting topological spaces at one point
JO  - Czechoslovak Mathematical Journal
PY  - 2006
SP  - 1193
EP  - 1206
VL  - 56
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a8/
LA  - en
ID  - CMJ_2006_56_4_a8
ER  - 
%0 Journal Article
%A Aliabad, Ali Rezaei
%T Pasting topological spaces at one point
%J Czechoslovak Mathematical Journal
%D 2006
%P 1193-1206
%V 56
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a8/
%G en
%F CMJ_2006_56_4_a8
Aliabad, Ali Rezaei. Pasting topological spaces at one point. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1193-1206. http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a8/

[1] A. R. Aliabad: $z^\circ $-ideals in  $C(X)$. PhD. Thesis, 1996.

[2] F. Azarpanah, O. A. S. Karamzadeh, and A.  Rezaei Aliabad: On ideals consisting entirely of zero divisors. Comm. Algebra 28 (2000), 1061–1073. | DOI | MR

[3] F.  Azarpanah, O, A. S.  Karamzadeh, and A.  Rezaei Aliabad: On $z^o-ideals$ in  $C(X)$. Fundamenta Math. 160 (1999), 15–25. | MR

[4] F. Azarpanah, O. A. S.  Karamzadeh: Algebraic characterizations of some disconnected spaces. Italian  J.  Pure Appl. Math. 10 (2001), 9–20. | MR

[5] R. Engelking: General Topology. PWN—Polish Scientific Publishing, , 1977. | MR | Zbl

[6] A. A. Estaji, O, A. S.  Karamzadeh: On $C(X)$ modulo its socle. Comm. Algebra 31 (2003), 1561–1571. | DOI | MR

[7] L. Gillman, M.  Jerison: Rings of Continuous Functions. Van Nostrand Reinhold, New York, 1960. | MR

[8] M. Henriksen, R. G. Wilson: Almost discrete $SV$-space. Topology and its Application 46 (1992), 89–97. | MR

[9] M. Henriksen, S.  Larson, J. Martinez, and R. G. Woods: Lattice-ordered algebras that are subdirect products of valuation domains. Trans. Amer. Math. Soc. 345 (1994), 195–221. | DOI | MR

[10] O. A. S.  Karamzadeh, M.  Rostami: On the intrinsic topology and some related ideals of  $C(X)$. Proc. Amer. Math. Soc. 93 (1985), 179–184. | MR

[11] S. Larson: $f$-rings in which every maximal ideal contains finitely many prime ideals. Comm. Algebra 25 (1997), 3859–3888. | DOI | MR

[12] R. Levy: Almost $P$-spaces. Can. J.  Math. 2 (1977), 284–288. | MR | Zbl

[13] S. Willard: General Topology. Addison Wesley, Reading, 1970. | MR | Zbl