Mathematics

In this paper, we obtain some sufficient conditions for the D-completion of a T0 space to be the well-filterification of this space, the well-filterification of a T0 space to be the sobrification of this space and the D-completion of a T0 space to be the sobrification, respectively. Moreover, we give an example to show that a tapered closed set may be neither the closure of a directed set nor the closed KF-set, respectively. Because the tapered closed set is a closed WD-set, the example also gives a negative answer to a problem proposed by Xu. Meantime, a new direct characterization of the D-completion of a T0 space is given by using the notion of pre-c-compact elements.

Let X be a set, B X denotes the family of all subsets of X and F:X⟶ B X be a set-valued mapping such that x∈F(x) , su p x∈X |F(x)|<κ , su p x∈X | F −1 (x)|<κ for all x∈X and some infinite cardinal κ . Then there exists a family F of bijective selectors of F such that |F|<κ and F(x)={f(x):f∈F} for each x∈X . We apply this result to G -space representations of balleans.

For a given continuum X and a natural number n, we consider the hyperspace F n (X) of all nonempty subsets of X with at most n points, metrized by the Hausdorff metric. In this paper we show that if X is a dendrite whose set of end points is closed, n∈N and Y is a continuum such that the hyperspaces F n (X) and F n (Y) are homeomorphic, then Y is a dendrite whose set of end points is closed.

A denumerable cellular family of a topological space X is an infinitely countable collection of pairwise disjoint non-empty open sets of X . It is proved that the following statements are equivalent in ZF : (i) For every infinite set X,[X ] <ω has a denumerable subset. (ii) Every infinite 0 -dimensional Hausdorff space admits a denumerable cellular family. It is also proved that (i) implies the following: (iii) Every infinite Hausdorff Baire space has a denumerable cellular family. Among other results, the following theorems are also proved in ZF : (iv) Every countable collection of non-empty subsets of R has a choice function iff, for every infinite second-countable Hausdorff space X , it holds that every base of X contains a denumerable cellular family of X . (v) If every Cantor cube is pseudocompact, then every non-empty countable collection of non-empty finite sets has a choice function. (vi) If all Cantor cubes are countably paracompact, then (i) holds. Moreover, among other forms independent of ZF , a partial Kinna-Wagner selection principle for families expressible as countable unions of finite families of finite sets is introduced. It is proved that if this new selection principle and (i) hold, then every infinite Boolean algebra has a tower and every infinite Hausdorff space has a denumerable cellular family.

Let X be a Hausdorff space and let H be one of the hyperspaces CL(X) , K(X) , F(X) or F n (X) ( n a positive integer) with the Vietoris topology. We study the following disconnectedness properties for H : extremal disconnectedness, being a F ′ -space, P -space or weak P -space and hereditary disconnectedness. Our main result states: if X is Hausdorff and F⊂X is a closed subset such that (a) both F and X−F are totally disconnected, (b) the quotient X/F is hereditarily disconnected, then K(X) is hereditarily disconnected. We also show an example proving that this result cannot be reversed.

This paper gives a process for finding discrete real specializations of sesquilinear representations of the braid groups using Salem numbers. This method is applied to the Jones and BMW representations, and some details on the commensurability of the target groups are given.

We construct two connected plane sets which can be embedded into rational curves. The first is a biconnected set with a dispersion point. It answers a question of Joachim Grispolakis. The second is indecomposable. Both examples are completely metrizable.

We take the abstract basis approach to classical domain theory and extend it to quantitative domains. In doing so, we provide dual characterisations of distance domains (some new even in the classical case) as well as unifying and extending previous formal ball dualities, namely the Kostanek-Waszkiewicz and Romaguero-Valero theorems. In passing, we also characterise hemimetric spaces that admit a hemimetric Smyth completion.

We study G δ subspaces of continuous dcpos, which we call domain-complete spaces, and G δ subspaces of locally compact sober spaces, which we call LCS-complete spaces. Those include all locally compact sober spaces-in particular, all continuous dcpos-, all topologically complete spaces in the sense of Čech, and all quasi-Polish spaces-in particular, all Polish spaces. We show that LCS-complete spaces are sober, Wilker, compactly Choquet-complete, completely Baire, and ⊙ -consonant-in particular, consonant; that the countably-based LCS-complete (resp., domain-complete) spaces are the quasi-Polish spaces exactly; and that the metrizable LCS-complete (resp., domain-complete) spaces are the completely metrizable spaces. We include two applications: on LCS-complete spaces, all continuous valuations extend to measures, and sublinear previsions form a space homeomorphic to the convex Hoare powerdomain of the space of continuous valuations.

We call a pair of infinite cardinals (κ,λ) with κ>λ a dominating (resp. pinning down) pair for a topological space X if for every subset A of X (resp. family U of non-empty open sets in X ) of cardinality ≤κ there is B⊂X of cardinality ≤λ such that A⊂ B ¯ ¯ ¯ ¯ (resp. B∩U≠∅ for each U∈U ). Clearly, a dominating pair is also a pinning down pair for X . Our definitions generalize the concepts introduced in [GTW] resp. [BT] which focused on pairs of the form ( 2 λ ,λ) . The main aim of this paper is to answer a large number of the numerous problems from [GTW] and [BT] that asked if certain conditions on a space X together with the assumption that ( 2 λ ,λ) or (( 2 λ ) + ,λ) is a pinning down pair or \dominating pair for X would imply d(X)≤λ . [BT] A. Bella, V.V. Tkachuk, Exponential density vs exponential domination, preprint [GTW] G. Gruenhage, V.V. Tkachuk, R.G. Wilson, Domination by small sets versus density, Topology and its Applications 282 (2020)