SP-scattered spaces; a new generalization of scattered spaces

Melvin Henriksen; Robert M. Raphael; Grant R. Woods

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 3, page 487-505
  • ISSN: 0010-2628

Abstract

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The set of isolated points (resp. P -points) of a Tychonoff space X is denoted by Is ( X ) (resp. P ( X ) ) . Recall that X is said to be scattered if Is ( A ) ≠ ⌀ whenever ⌀ ≠ A ⊂ X . If instead we require only that P ( A ) has nonempty interior whenever ⌀ ≠ A ⊂ X , we say that X is SP-scattered. Many theorems about scattered spaces hold or have analogs for SP-scattered spaces. For example, the union of a locally finite collection of SP-scattered spaces is SP-scattered. Some known theorems about Lindelöf or paracompact scattered spaces hold also in case the spaces are SP-scattered. If X is a Lindelöf or a paracompact SP-scattered space, then so is its P -coreflection. Some results are given on when the product of two Lindelöf or paracompact spaces is Lindelöf or paracompact when at least one of the factors is SP-scattered. We relate our results to some on RG-spaces and z -dimension.

How to cite

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Henriksen, Melvin, Raphael, Robert M., and Woods, Grant R.. "SP-scattered spaces; a new generalization of scattered spaces." Commentationes Mathematicae Universitatis Carolinae 48.3 (2007): 487-505. <http://eudml.org/doc/250219>.

@article{Henriksen2007,
abstract = {The set of isolated points (resp. $P$-points) of a Tychonoff space $X$ is denoted by $\operatorname\{Is\}(X)$ (resp. $P(X))$. Recall that $X$ is said to be scattered if $\operatorname\{Is\}(A)\ne \varnothing $ whenever $\varnothing \ne A\subset X$. If instead we require only that $P(A)$ has nonempty interior whenever $\varnothing \ne A\subset X$, we say that $X$ is SP-scattered. Many theorems about scattered spaces hold or have analogs for SP-scattered spaces. For example, the union of a locally finite collection of SP-scattered spaces is SP-scattered. Some known theorems about Lindelöf or paracompact scattered spaces hold also in case the spaces are SP-scattered. If $X$ is a Lindelöf or a paracompact SP-scattered space, then so is its $P$-coreflection. Some results are given on when the product of two Lindelöf or paracompact spaces is Lindelöf or paracompact when at least one of the factors is SP-scattered. We relate our results to some on RG-spaces and $z$-dimension.},
author = {Henriksen, Melvin, Raphael, Robert M., Woods, Grant R.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {scattered spaces; SP-scattered spaces; CB-index; sp-index; $P$-points; $P$-spaces; strong $P$-points; RG-spaces; $z$-dimension; locally finite; Lindelöf spaces; paracompact spaces; $P$-coreflection; $G_\{\delta \}$-topology; product spaces; scattered spaces; SP-scattered spaces; CB-index; sp-index; -points; -spaces; strong -points},
language = {eng},
number = {3},
pages = {487-505},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {SP-scattered spaces; a new generalization of scattered spaces},
url = {http://eudml.org/doc/250219},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Henriksen, Melvin
AU - Raphael, Robert M.
AU - Woods, Grant R.
TI - SP-scattered spaces; a new generalization of scattered spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 3
SP - 487
EP - 505
AB - The set of isolated points (resp. $P$-points) of a Tychonoff space $X$ is denoted by $\operatorname{Is}(X)$ (resp. $P(X))$. Recall that $X$ is said to be scattered if $\operatorname{Is}(A)\ne \varnothing $ whenever $\varnothing \ne A\subset X$. If instead we require only that $P(A)$ has nonempty interior whenever $\varnothing \ne A\subset X$, we say that $X$ is SP-scattered. Many theorems about scattered spaces hold or have analogs for SP-scattered spaces. For example, the union of a locally finite collection of SP-scattered spaces is SP-scattered. Some known theorems about Lindelöf or paracompact scattered spaces hold also in case the spaces are SP-scattered. If $X$ is a Lindelöf or a paracompact SP-scattered space, then so is its $P$-coreflection. Some results are given on when the product of two Lindelöf or paracompact spaces is Lindelöf or paracompact when at least one of the factors is SP-scattered. We relate our results to some on RG-spaces and $z$-dimension.
LA - eng
KW - scattered spaces; SP-scattered spaces; CB-index; sp-index; $P$-points; $P$-spaces; strong $P$-points; RG-spaces; $z$-dimension; locally finite; Lindelöf spaces; paracompact spaces; $P$-coreflection; $G_{\delta }$-topology; product spaces; scattered spaces; SP-scattered spaces; CB-index; sp-index; -points; -spaces; strong -points
UR - http://eudml.org/doc/250219
ER -

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