Mathematics

In this paper, we define asymptotic dimension of fuzzy metric spaces in the sense of George and Veeramini. We prove that asymptotic dimension is an invariant in the coarse category of fuzzy metric spaces. We also show several consequences of asymptotic dimension in the fuzzy setting which resemble the consequences of asymptotic dimension in the metric setting. We finish with calculating asymptotic dimension of a few fuzzy metric spaces.

We define the direct sum of a countable family of pointed metric spaces in a way resembling the direct sum of groups. Then we prove that if a family consists of 0 -hyperbolic (in the sense of Gromov) and D -discrete spaces, then its direct sum has asymptotic property C. The main example is a countable direct sum of free groups of (possibly varying) finite rank. This is a generalization of T. Yamauchi's result concernig the countable direct sum of the integers.

For a Tychonoff space X and a subspace Y⊂R , we study Baire category properties of the space C ↓F (X,Y) of continuous functions from X to Y , endowed with the Fell hypograph topology. We characterize pairs X,Y for which the function space C ↓F (X,Y) is ∞ -meager, meager, Baire, Choquet, strong Choquet, (almost) complete-metrizable or (almost) Polish.

We initiate a systematic investigation of group actions on compact medain algebras via the corresponding dynamics on their spaces of measures. We show that a probability measure which is invariant under a natural push forward operation must be a uniform measure on a cube and use this to show that every amenable group action on a locally convex compact median algebra fixed a sub-cube.

A ballean B (or a coarse structure) on a set X is a family of subsets of X called balls (or entourages of the diagonal in X×X ) defined in such a way that B can be considered as the asymptotic counterpart of a uniform topological space. The aim of this paper is to study two concrete balleans defined by the ideals in the Boolean algebra of all subsets of X and their hyperballeans, with particular emphasis on their connectedness structure, more specifically the number of their connected components.

A topological space is defined to be banalytic (resp. analytic) if it is the image of a Polish space under a Borel (resp. continuous) map. A regular topological space is analytic if and only if it is banalytic and cosmic. Each (regular) banalytic space has countable spread (and under PFA is hereditarily Lindelöf). Applying banalytic spaces to topological groups, we prove that for a Baire topological group X the following conditions are equivalent: (1) X is Polish, (2) X is analytic, (3) X is banalytic and cosmic, (4) X is banalytic and has countable pseudocharacter. Under PFA the conditions (1)--(4) are equivalent to the banalycity of X . The conditions (1)--(3) remain equivalent for any Baire semitopological group.

In this paper, we introduce the notion of topologically Banach contraction mapping defined on an arbitrary topological space X with the help of a continuous function g:X×X→R and investigate the existence of fixed points of such mapping. Moreover, we introduce two types of mappings defined on a non-empty subset of X and produce sufficient conditions which will ensure the existence of best proximity points for these mappings. Our best proximity point results also extend some existing results from metric spaces or Banach spaces to topological spaces. More precisely, our newly introduced mappings are more general than that of the corresponding notions introduced by Bunlue and Suantai [Arch. Math. (Brno), 54(2018), 165-176]. We present several examples to validate our results and justify its motivation. To study best proximity point results, we introduce the notions of g-closed, g-sequentially compact subsets of X and produce examples to show that there exists a non-empty subset of X which is not closed, sequentially compact under usual topology but is g-closed and g-sequentially compact.

Call a compact space X pin homogeneous if every two points a,b are pin equivalent, meaning that there exists a compact space Y , a quotient map f:Y→X , and a homeomorphism g:Y→Y such that g f −1 {a}= f −1 {b} . We will prove a representation theorem for pin equivalence; transitivity of pin equivalence will be a corollary. Pin homogeneity is strictly weaker than homogeneity and pin equivalence is strictly stronger than Tukey equivalence. Just as with topological homogeneity, no infinite compact F -space is pin homogeneous. On the other hand, X× 2 χ(X) is pin homogeneous for every compact X . And there is a compact pin homogeneous space with points of different π -character.

We investigate when the idempotent barycenter map restricted to the points with no-trivial fibers is a trivial bundle with the fiber Hilbert cube.

We study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of Hall and Kulatilaka and a characterization of a certain class of Lie groups, due to Grosser and Herfort, we prove that a c-minimal locally solvable Lie group is compact. It is shown that if a topological group G contains a compact open normal subgroup N , then G is c-totally minimal if and only if G/N is hereditarily non-topologizable. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact.