Spaces X in which all prime z -ideals of C ( X ) are minimal or maximal

Melvin Henriksen; Jorge Martinez; Grant R. Woods

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 2, page 261-294
  • ISSN: 0010-2628

Abstract

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Quasi P -spaces are defined to be those Tychonoff spaces X such that each prime z -ideal of C ( X ) is either minimal or maximal. This article is devoted to a systematic study of these spaces, which are an obvious generalization of P -spaces. The compact quasi P -spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. A thorough account of locally compact quasi P -spaces is given. If X is a cozero-complemented space and every nowhere dense zeroset is a z -embedded P -space, then X is a quasi P -space. Conversely, if X is a quasi P -space and F is a nowhere dense z -embedded zeroset, then F is a P -space. On the other hand, there are examples of countable quasi P -spaces with no P -points at all. If a product X × Y is normal and quasi P , then one of the factors must be a P -space. Conversely, if one of the factors is a compact quasi P -space and the other a P -space then the product is quasi P . If X is normal and X and Y are cozero-complemented spaces and f : X ⟶ Y is a closed continuous surjection which has the property that f - 1 ( Z ) is nowhere dense for each nowhere dense zeroset Z , then if X is quasi P , so is Y . The converse fails even with more stringent assumptions on the map f . The paper then closes with a number of open questions, amongst which the most glaring is whether the free union of quasi P -spaces is always quasi P .

How to cite

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Henriksen, Melvin, Martinez, Jorge, and Woods, Grant R.. "Spaces $X$ in which all prime $z$-ideals of $C(X)$ are minimal or maximal." Commentationes Mathematicae Universitatis Carolinae 44.2 (2003): 261-294. <http://eudml.org/doc/249184>.

@article{Henriksen2003,
abstract = {Quasi $P$-spaces are defined to be those Tychonoff spaces $X$ such that each prime $z$-ideal of $C(X)$ is either minimal or maximal. This article is devoted to a systematic study of these spaces, which are an obvious generalization of $P$-spaces. The compact quasi $P$-spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. A thorough account of locally compact quasi $P$-spaces is given. If $X$ is a cozero-complemented space and every nowhere dense zeroset is a $z$-embedded $P$-space, then $X$ is a quasi $P$-space. Conversely, if $X$ is a quasi $P$-space and $F$ is a nowhere dense $z$-embedded zeroset, then $F$ is a $P$-space. On the other hand, there are examples of countable quasi $P$-spaces with no $P$-points at all. If a product $X\times Y$ is normal and quasi $P$, then one of the factors must be a $P$-space. Conversely, if one of the factors is a compact quasi $P$-space and the other a $P$-space then the product is quasi $P$. If $X$ is normal and $X$ and $Y$ are cozero-complemented spaces and $f:X\longrightarrow Y$ is a closed continuous surjection which has the property that $f^\{-1\}(Z)$ is nowhere dense for each nowhere dense zeroset $Z$, then if $X$ is quasi $P$, so is $Y$. The converse fails even with more stringent assumptions on the map $f$. The paper then closes with a number of open questions, amongst which the most glaring is whether the free union of quasi $P$-spaces is always quasi $P$.},
author = {Henriksen, Melvin, Martinez, Jorge, Woods, Grant R.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasi $P$-space; $P$-space; scattered space; Cantor-Bendixson derivatives; nodec space; quasinormality; quasi -space; -space; scattered space; Cantor-Bendixson derivatives; nodec space; quasinormality},
language = {eng},
number = {2},
pages = {261-294},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Spaces $X$ in which all prime $z$-ideals of $C(X)$ are minimal or maximal},
url = {http://eudml.org/doc/249184},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Henriksen, Melvin
AU - Martinez, Jorge
AU - Woods, Grant R.
TI - Spaces $X$ in which all prime $z$-ideals of $C(X)$ are minimal or maximal
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 2
SP - 261
EP - 294
AB - Quasi $P$-spaces are defined to be those Tychonoff spaces $X$ such that each prime $z$-ideal of $C(X)$ is either minimal or maximal. This article is devoted to a systematic study of these spaces, which are an obvious generalization of $P$-spaces. The compact quasi $P$-spaces are characterized as the compact spaces which are scattered and of Cantor-Bendixson index no greater than 2. A thorough account of locally compact quasi $P$-spaces is given. If $X$ is a cozero-complemented space and every nowhere dense zeroset is a $z$-embedded $P$-space, then $X$ is a quasi $P$-space. Conversely, if $X$ is a quasi $P$-space and $F$ is a nowhere dense $z$-embedded zeroset, then $F$ is a $P$-space. On the other hand, there are examples of countable quasi $P$-spaces with no $P$-points at all. If a product $X\times Y$ is normal and quasi $P$, then one of the factors must be a $P$-space. Conversely, if one of the factors is a compact quasi $P$-space and the other a $P$-space then the product is quasi $P$. If $X$ is normal and $X$ and $Y$ are cozero-complemented spaces and $f:X\longrightarrow Y$ is a closed continuous surjection which has the property that $f^{-1}(Z)$ is nowhere dense for each nowhere dense zeroset $Z$, then if $X$ is quasi $P$, so is $Y$. The converse fails even with more stringent assumptions on the map $f$. The paper then closes with a number of open questions, amongst which the most glaring is whether the free union of quasi $P$-spaces is always quasi $P$.
LA - eng
KW - quasi $P$-space; $P$-space; scattered space; Cantor-Bendixson derivatives; nodec space; quasinormality; quasi -space; -space; scattered space; Cantor-Bendixson derivatives; nodec space; quasinormality
UR - http://eudml.org/doc/249184
ER -

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