Mathematics

Often, a given selection game studied in the literature has a known dual game. In dual games, a winning strategy for a player in either game may be used to create a winning strategy for the opponent in the dual. For example, the Rothberger selection game involving open covers is dual to the point-open game. This extends to a general theorem: if {ranf:f∈C(R)} is coinitial in A with respect to ⊆ , where C(R)={f∈(⋃R ) R :R∈R⇒f(R)∈R} collects the choice functions on the set R , then G 1 (A,B) and G 1 (R,¬B) are dual selection games.

We modify our previous construction of link homology in order to include a natural duality functor F . To a link L we associate a triply-graded module HXY(L) over the graded polynomial ring R(L)=C[ x 1 , y 1 ,…, x ℓ , y ℓ ] . The module has an involution F that intertwines the Fourier transform on R(L) , F( x i )= y i , F( y i )= x i . In the case when ℓ=1 the module is free over R(L) and specialization to x=y=0 matches with the triply-graded knot homology previously constructed by the authors. Thus we show that the corresponding super-polynomial satisfies the categorical version of q→1/q symmetry. We also construct an isotopy invariant of the closure of a dichromatic braid and relate this invariant to HXY(L) .

A topological group X is called duoseparable if there exists a countable set S⊆X such that SUS=X for any neighborhood U⊆X of the unit. We construct a functor F assigning to each (abelian) topological group X a duoseparable (abelain-by-cyclic) topological group FX , containing an isomorphic copy of X . In fact, the functor F is defined on the category of unital topologized magmas. Also we prove that each σ -compact locally compact abelian topological group embeds into a duoseparable locally compact abelian-by-countable topological group.

Let κ be an infinite cardinal. A topological space X is κ -bounded if the closure of any subset of cardinality ≤κ in X is compact. We discuss the problem of embeddability of topological spaces into Hausdorff (Urysohn, regular) κ -bounded spaces, and present a canonical construction of such an embedding. Also we construct a (consistent) example of a sequentially compact separable regular space that cannot be embedded into a Hausdorff ω -bounded space.

In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example of a regular separable scattered topological space which cannot be embedded into an Urysohn countably compact topological space but embeds into a Hausdorff countably compact space.

We initiate the study of ends of non-metrizable manifolds and introduce the notion of short and long ends. Using the theory developed, we provide a characterization of (non-metrizable) surfaces that can be written as the topological sum of a metrizable manifold plus a countable number of "long pipes" in terms of their spaces of ends; this is a direct generalization of Nyikos's bagpipe theorem.

The enumeration degrees of sets of natural numbers can be identified with the degrees of difficulty of enumerating neighborhood bases of points in a universal second-countable T 0 -space (e.g. the ω -power of the Sierpiński space). Hence, every represented second-countable T 0 -space determines a collection of enumeration degrees. For instance, Cantor space captures the total degrees, and the Hilbert cube captures the continuous degrees by definition. Based on these observations, we utilize general topology (particularly non-metrizable topology) to establish a classification theory of enumeration degrees of sets of natural numbers.

If X is a finite tree and f:X⟶X is a map, as the Main Theorem of this paper we find eight conditions, each of which is equivalent to the fact that f is equicontinuous. To name just a few of the results obtained: the equicontinuity of f is equivalent to the fact that there is no arc A⊆X satisfying A⊊ f n [A] for some n∈N . It is also equivalent to the fact that for some nonprincial ultrafilter u , the function f u :X⟶X is continuous (in other words, failure of equicontinuity of f is equivalent to the failure of continuity of every element of the Ellis remainder g∈E(X,f ) ∗ ). One of the tools used in the proofs is the Ramsey-theoretic result known as Hindman's theorem. Our results generalize the ones shown by Vidal-Escobar and García-Ferreira, and complement those of Bruckner and Ceder, Mai, and Camargo, Rincón and Uzcátegui.

This paper studies n-player games where players beliefs about their opponents behaviour are capacities (fuzzy measures, non-additive probabilities). The concept of an equilibrium under uncertainty was introduced by J.Dow and S.Werlang (1994) for two players and was extended to n-player games by J.Eichberger and D.Kelsey (2000). Expected utility (payoff function) was expressed by Choquet integral. The concept of an equilibrium under uncertainty with expected utility expressed by Sugeno integral were considered by T.Radul (2018). We consider in this paper an equilibrium with expected utility expressed by fuzzy integral generated by a continuous t-norm which is a natural generalization of Sugeno integral.

We estimate size of recurrence of an action of a nilpotent group by homeomorphisms of a compact space for polynomial mappings into a nilpotent group form the partial semigroup ( P f (N),⊎) . To do this we have used algebraic structure of the Stone-Čech copactification partial semigroup and that of the given nilpotent group.