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.
© 2007 Charles University in Prague
Henriksen, Melvin, R. Raphael, and R. G. Woods. "SP-scattered spaces: a new generalization of scattered spaces." Commentationes Mathematicae Universitatis Carolinae 48.3 (2007): 487-505.