Pasting topological spaces at one point

Ali Rezaei Aliabad

Czechoslovak Mathematical Journal (2006)

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

Abstract

top
Let { X α } α ∈ Λ be a family of topological spaces and x α ∈ X α , for every α ∈ Λ . Suppose X is the quotient space of the disjoint union of X α ’s by identifying x α ’s as one point σ . We try to characterize ideals of C ( X ) according to the same ideals of C ( X α ) ’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].

How to cite

top

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

top
  1. z ∘ -ideals in  C ( X ) , PhD. Thesis, 1996. (1996) 
  2. 10.1080/00927870008826878, Comm. Algebra 28 (2000), 1061–1073. (2000) MR1736781DOI10.1080/00927870008826878
  3. On z o - i d e a l s in  C ( X ) , Fundamenta Math. 160 (1999), 15–25. (1999) MR1694400
  4. Algebraic characterizations of some disconnected spaces, Italian  J.  Pure Appl. Math. 10 (2001), 9–20. (2001) MR1962109
  5. General Topology, PWN—Polish Scientific Publishing, , 1977. (1977) Zbl0373.54002MR0500780
  6. 10.1081/AGB-120018497, Comm. Algebra 31 (2003), 1561–1571. (2003) MR1972881DOI10.1081/AGB-120018497
  7. Rings of Continuous Functions, Van Nostrand Reinhold, New York, 1960. (1960) MR0116199
  8. Almost discrete S V -space, Topology and its Application 46 (1992), 89–97. (1992) MR1184107
  9. 10.1090/S0002-9947-1994-1239640-0, Trans. Amer. Math. Soc. 345 (1994), 195–221. (1994) MR1239640DOI10.1090/S0002-9947-1994-1239640-0
  10. On the intrinsic topology and some related ideals of  C ( X ) , Proc. Amer. Math. Soc. 93 (1985), 179–184. (1985) MR0766552
  11. 10.1080/00927879708826092, Comm. Algebra 25 (1997), 3859–3888. (1997) MR1481572DOI10.1080/00927879708826092
  12. Almost P -spaces, Can. J.  Math. 2 (1977), 284–288. (1977) Zbl0342.54032MR0464203
  13. General Topology, Addison Wesley, Reading, 1970. (1970) Zbl0205.26601MR0264581

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.