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As usual will denote the ring of real-valued continuous functions on a Tychonoff space . It is well-known that if and are realcompact spaces such that and are isomorphic, then and are homeomorphic; that is determines. The restriction to realcompact spaces stems from the fact that and are isomorphic, where is the (Hewitt) realcompactification of . In this note, a class of locally compact spaces that includes properly the class of locally compact realcompact spaces is exhibited...
For a space Z let 𝒦(Z) denote the partially ordered set of all compact subspaces of Z under set inclusion. If X is a compact space, Δ is the diagonal in X², and 𝒦(X²∖Δ) has calibre (ω₁,ω), then X is metrizable. There is a compact space X such that X²∖Δ has relative calibre (ω₁,ω) in 𝒦(X²∖Δ), but which is not metrizable. Questions of Cascales et al. (2011) concerning order constraints on 𝒦(A) for every subspace of a space X are answered.
A hull of A ⊆ [0,1] is a set H containing A such that λ*(H) = λ*(A). We investigate all four versions of the following problem. Does there exist a monotone (with respect to inclusion) map that assigns a Borel/ hull to every negligible/measurable subset of [0,1]?
Three versions turn out to be independent of ZFC, while in the fourth case we only prove that the nonexistence of a monotone hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent....
For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space such that , and no closed subset L of with ind L less than the predecessor of α is a partition in . An α-dimensional Cantor Ind-manifold can be constructed similarly.
Following Preuss' general connectedness theory in topological categories, a connectedness concept for approach spaces is introduced, which unifies topological connectedness in the setting of topological spaces, and Cantor-connectedness in the setting of metric spaces.
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