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In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse structures on $omega$, about the interplay between properties of a coarse space and its Higson corona, about some special ultrafilters ($T$-points and cellular $T$-points) related to finitary coarse structures on $omega$, about partitions of coarse spaces into thin pieces, and also about coarse groups having some extremal properties.
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Werners set-theoretical model is one of the simplest models of CIC. It combines a functional view of predicative universes with a collapsed view of the impredicative sort Prop. However this model of Prop is so coarse that the principle of excluded middle holds. Following our previous work, we interpret Prop into a topological space (a special case of Heyting algebra) to make the model more intuitionistic without sacrificing simplicity. We improve on that work by providing a full interpretation of dependent product types, using Alexandroff spaces. We also extend our approach to inductive types by adding support for lists.
Which Isbell--Mrowka spaces ($Psi$-spaces) satisfy the star version of Mengers and Hurewiczs covering properties? Following Bonanzinga and Matveev, this question is considered here from a combinatorial point of view. An example of a $Psi$-space that is (strongly) star-Menger but not star-Hurewicz is obtained. The PCF-theory function $kappamapstocof([kappa]^alephes)$ is a key tool. Using the method of forcing, a complete answer to a question of Bonanzinga and Matveev is provided. The results also apply to the mentioned covering properties in the realm of Pixley--Roy spaces, to the extent of spaces with these properties, and to the character of free abelian topological groups over hemicompact $k$ spaces.
We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of omega_1 is hereditarily paracompact.
We examine locally compact normal spaces in models of form PFA(S)[S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of omega_1 and in which all separable closed subspaces are Lindelof.
Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.
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