| Title:
|
SP-scattered spaces; a new generalization of scattered spaces (English) |
| Author:
|
Henriksen, M. |
| Author:
|
Raphael, R. |
| Author:
|
Woods, R. G. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
3 |
| Year:
|
2007 |
| Pages:
|
487-505 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 {\it scattered\/} if $\operatorname{Is}(A)\neq \varnothing $ whenever $\varnothing \neq A\subset X$. If instead we require only that $P(A)$ has nonempty interior whenever $\varnothing \neq A\subset X$, we say that $X$ is {\it SP-scattered\/}. Many theorems about scattered spaces hold or have analogs for {\it 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. (English) |
| Keyword:
|
scattered spaces |
| Keyword:
|
SP-scattered spaces |
| Keyword:
|
CB-index |
| Keyword:
|
sp-index |
| Keyword:
|
$P$-points |
| Keyword:
|
$P$-spaces |
| Keyword:
|
strong $P$-points |
| Keyword:
|
RG-spaces |
| Keyword:
|
$z$-dimension |
| Keyword:
|
locally finite |
| Keyword:
|
Lindelöf spaces |
| Keyword:
|
paracompact spaces |
| Keyword:
|
$P$-coreflection |
| Keyword:
|
$G_{\delta}$-topology |
| Keyword:
|
product spaces |
| MSC:
|
54G10 |
| MSC:
|
54G12 |
| idZBL:
|
Zbl 1199.54188 |
| idMR:
|
MR2374129 |
| . |
| Date available:
|
2009-05-05T17:04:16Z |
| Last updated:
|
2012-05-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119674 |
| . |
| Reference:
|
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| . |
