M. Henriksen, R. Raphael, R.G. Woods
SP-scattered spaces; a new generalization of scattered spaces

Comment.Math.Univ.Carolin. 48,3 (2007) 487-505.

Abstract: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)\not =\varnothing $ whenever $\varnothing \not =A\subset X$. If instead we require only that $P(A)$ has nonempty interior whenever $\varnothing \not =A\subset X$, 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 Lindel\"of or paracompact scattered spaces hold also in case the spaces are SP-scattered. If $X$ is a Lindel\"of or a paracompact SP-scattered space, then so is its $P$-coreflection. Some results are given on when the product of two Lindel\"of or paracompact spaces is Lindel\"of or paracompact when at least one of the factors is SP-scattered. We relate our results to some on RG-spaces and $z$-dimension.

Keywords: scattered spaces, SP-scattered spaces, CB-index, sp-index, $P$-points, $P$-spaces, strong $P$-points, RG-spaces, $z$-dimension, locally finite, Lindel\"of spaces, paracompact spaces, $P$-coreflection, $G_{\delta }$-topology, product spaces
AMS Subject Classification: 54G10, 54G12