SP-Scattered Spaces: A New Generalization of Scattered Spaces

Authors

Melvin Henriksen, Harvey Mudd CollegeFollow
Robert M. Raphael, Concordia University
R. G. Woods, University of Manitoba

Document Type

Article

Department

Mathematics (HMC)

Publication Date

2007

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) ≠ ∅ 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.

Comments

Previously linked to as: http://ccdl.libraries.claremont.edu/u?/irw,433.

Publisher pdf, posted with permission.

Article can also be found at http://dml.cz/dmlcz/119674

Rights Information

© 2007 Charles University in Prague

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Recommended Citation

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.