SP-scattered spaces; a new generalization of scattered spaces

Henriksen, M. ; Raphael, R. ; Woods, R. G.

Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 3, pp. 487-505.

Résumé

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.

Détail

Classification : 54G10, 54G12
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