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SP-scattered spaces; a new generalization of scattered spaces

Melvin HenriksenRobert M. RaphaelGrant R. Woods — 2007

Commentationes Mathematicae Universitatis Carolinae

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 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 . Many theorems about scattered spaces hold or have analogs for 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....

Contractible simplicial objects

Michael BarrJohn F. KennisonRobert M. Raphael — 2019

Commentationes Mathematicae Universitatis Carolinae

We raise the question of when a simplicial object in a catetgory is deemed contractible. The literature offers three definitions. One is the existence of an “extra degeneracy”, indexed by - 1 , which does not quite live up to the name. This can be strengthened to a “strong extra degeneracy". Another possibility is that it be homotopic to a constant simplicial object. Despite claims in the literature to the contrary, we show that all three are distinct concepts with strong extra degeneracy implies extra...

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