On nonregular ideals and z ∘ -ideals in C ( X )

F. Azarpanah; M. Karavan

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 2, page 397-407
  • ISSN: 0011-4642

Abstract

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The spaces X in which every prime z ∘ -ideal of C ( X ) is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces X , such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime z ∘ -ideal in C ( X ) is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in C ( X ) a z ∘ -ideal? When is every nonregular (prime) z -ideal in C ( X ) a z ∘ -ideal? For instance, we show that every nonregular prime ideal of C ( X ) is a z ∘ -ideal if and only if X is a ∂ -space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior).

How to cite

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Azarpanah, F., and Karavan, M.. "On nonregular ideals and $z^\circ $-ideals in $C(X)$." Czechoslovak Mathematical Journal 55.2 (2005): 397-407. <http://eudml.org/doc/30953>.

@article{Azarpanah2005,
abstract = {The spaces $X$ in which every prime $z^\circ $-ideal of $C(X)$ is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces $X$, such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime $z^\circ $-ideal in $C(X)$ is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in $C(X)$ a $z^\circ $-ideal? When is every nonregular (prime) $z$-ideal in $C(X)$ a $z^\circ $-ideal? For instance, we show that every nonregular prime ideal of $C(X)$ is a $z^\circ $-ideal if and only if $X$ is a $\partial $-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior).},
author = {Azarpanah, F., Karavan, M.},
journal = {Czechoslovak Mathematical Journal},
keywords = {$z^\circ $-ideal; prime $z$-ideal; nonregular ideal; almost $\{P\}$-space; $\partial $-space; $m$-space; -ideal; prime -ideal; nonregular ideal; almost -space; -space; -space},
language = {eng},
number = {2},
pages = {397-407},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On nonregular ideals and $z^\circ $-ideals in $C(X)$},
url = {http://eudml.org/doc/30953},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Azarpanah, F.
AU - Karavan, M.
TI - On nonregular ideals and $z^\circ $-ideals in $C(X)$
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 2
SP - 397
EP - 407
AB - The spaces $X$ in which every prime $z^\circ $-ideal of $C(X)$ is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces $X$, such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime $z^\circ $-ideal in $C(X)$ is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in $C(X)$ a $z^\circ $-ideal? When is every nonregular (prime) $z$-ideal in $C(X)$ a $z^\circ $-ideal? For instance, we show that every nonregular prime ideal of $C(X)$ is a $z^\circ $-ideal if and only if $X$ is a $\partial $-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior).
LA - eng
KW - $z^\circ $-ideal; prime $z$-ideal; nonregular ideal; almost ${P}$-space; $\partial $-space; $m$-space; -ideal; prime -ideal; nonregular ideal; almost -space; -space; -space
UR - http://eudml.org/doc/30953
ER -

References

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  10. 10.4153/CJM-1977-030-7, Canad. J.  Math. 2 (1977), 284–288. (1977) Zbl0342.54032MR0464203DOI10.4153/CJM-1977-030-7
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