Order completions of semiprime rings
SP-scattered spaces; a new generalization of scattered spaces
The set of isolated points (resp. -points) of a Tychonoff space is denoted by (resp. . Recall that is said to be if whenever . If instead we require only that has nonempty interior whenever , we say that 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
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 , 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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