
GonzálezSilva, R.A.; Hrusák, M.
[EN] Two different openpoint games are studied here, the Ggame and the Gpgame, defined for each p ∈ ω∗. We prove that for each p ∈ ω∗, there exists a space in which none of the players of the Gpgame has a winning strategy.Nevertheless a result of P. Nyikos, essentially shows that it is consistent, that there exists a countable space in which all these games are undetermined.We construct a countably compact space in which player II of the Gpgame is the winner, for every p ∈ ω∗. With the same technique of construction we built a countably compact space...

Beer, Gerald; Segura, Manuel
[EN] Given a continuous nonnegative functional λ that makes sense defined on an arbitrary metric space (X, d), one may consider those spaces in which each sequence (xn) for which lim n→∞λ(xn) = 0 clusters. The compact metric spaces, the complete metric spaces, the cofinally complete metric spaces, and the UCspaces all arise in this way. Starting with a general continuous nonnegative functional λ defined on (X, d), we study the bornology Bλ of all subsets A of X on which limn→∞λ(an) = 0 ⇒ (an) clusters, treating the possibility X ∈ Bλ as a special case. We characterize those...

Georgiou, D.N.
[EN] Let Y and Z be two fixed topological spaces, O(Z) the family of all open subsets of Z, C(Y,Z) the set of all continuous maps from Y to Z, and OZ(Y ) the set {f−1(U) : f ϵ C(Y,Z) and U ϵ O(Z)}. In this paper, we give and study new topologies on the sets C(Y,Z) and OZ(Y ) calling (A,A0)splitting and (A,A0)admissible, where A and A0 families of spaces.

Yang, Zhanbo
[EN] We first study subspaces and product spaces in the context of nearness spaces and prove that UN spaces, CN spaces, PN spaces and totally bounded nearness spaces are nearness hereditary; TN spaces and compact nearness spaces are Nclosed hereditary. We prove that N2 plus compact implies Nclosed subsets. We prove that totally bounded, compact and N2 are productive. We generalize the concepts of neighborhood systems into the nearness spaces and prove that the nearness neighborhood systems are consistent with existing concepts of neighborhood systems in topological spaces, uniform spaces and proximity spaces respectively when considered in the respective subcategories....

Kohli, J.K.; Singh, D.; Aggarwal, Jeetendra
[EN] A strong variant of continuity called ‘Fsupercontinuity’ is introduced. The class of Fsupercontinuous functions strictly contains the class of zsupercontinuous functions (Indian J. Pure Appl. Math. 33 (7) (2002), 1097–1108) which in turn properly contains the class of clsupercontinuous functions ( clopen maps) (Appl. Gen. Topology 8 (2) (2007), 293–300; Indian J. Pure Appl. Math. 14 (6) (1983), 762–772). Further, the class of Fsupercontinuous functions is properly contained in the class of Rsupercontinuous functions which in turn is strictly contained in the class of continuous functions. Basic properties of Fsupercontinuous functions are studied and their place in the hierarchy...

Song, YanKui
[EN] A space X is discretely absolutely starLindelöf if for every open cover U of X and every dense subset D of X, there exists a countable subset F of D such that F is discrete closed in X and St(F, U) = X, where St(F, U) = S{U ∈ U : U ∩F 6= Ø}. We show that every Hausdorff starLindelöf space can be represented in a Hausdorff discretely absolutely starLindelöf space as a closed Gsubspace.

Tkachuk, Vladimir V.
[EN] We show, in particular, that if nw(Nt) ≤ k for any t ϵ T and C is a dense subspace of the product ǁ{Nt : t 2 T} then, for any continuous (not necessarily surjective) map ϕ : C ͢ K of C into a compact space K with t(K) ≤ k, we have ψ(ϕ (C)) ≤ k. This result has several applications in Cptheory. We prove, among other things, that if K is a nonmetrizable Corson compact space then Cp(K) cannot be condensed onto a σcompact space. This answers two questions published by Arhangel’skii and Pavlov.

Sankar Raj, V.; Veeramani, P.
[EN] Let A, B be nonempty closed bounded convex subsets of a uniformly convex Banach space and T : A∪B → A∪B be a map such that T(A) ⊆ B, T(B) ⊆ A and ǁTx − Tyǁ ≤ ǁx − yǁ, for x in A and y in B. The fixed point equation Tx = x does not possess a solution when A ∩ B = Ø. In such a situation it is natural to explore to find an element x0 in A satisfying ǁx0 − Tx0ǁ = inf{ǁa − bǁ : a ∈ A, b ∈ B} = dist(A,B)....

Olela Otafudu, Olivier
[EN] We introduce and investigate the concept of geodesic bicombing in T0quasimetric spaces. We prove that any Isbellconvex T0quasimetric space admits a geodesic bicombing which satisfies the equivariance property. Additionally, we show that many results on geodesic bicombing hold in quasimetric settings, provided that non symmetry in quasimetric spaces holds.

Yan, PengFei; Hu, XingYu; Xie, LiHong
[EN] In this paper, the concept of kupper semicontinuous setvalued mappings is introduced. Using this concept, we give characterizations of ksemistratifiable and kMCM spaces, which answers a question posed by Xie and Yan.

Pant, Rajendra
[EN] We introduce a new type of nonlinear contraction and present some fixed point results without using continuity or semicontinuity. Our result complement, extend and generalize a number of fixed point theorems including the the wellknown Boyd and Wong theorem [On nonlinear contractions, Proc. Amer. Math. Soc. 20(1969)]. Also we discuss an application to iterated function systems.

Bella, Angelo
[EN] A pseudocompact space is maximal pseudocompact if every strictly finer topology is no longer pseudocompact. The main result here is a counterexample which answers a question rised by Alas, Sanchis and Wilson.

Jäger, Gunther; Ahsanullah, T. M. G.
[EN] We identify two categories of quantalevalued convergence tower spaces that are isomorphic to the categories of quantalevalued metric spaces and quantalevalued partial metric spaces, respectively. As an application we state a quantalevalued metrization theorem for quantalevalued convergence tower groups.

Rubin, Leonard R.
[EN] It has been shown by S. Mardešić that if a compact metrizable space X has dim X ≥ 1 and X is the inverse limit of an inverse sequence of compact triangulated polyhedra with simplicial bonding maps, then X must contain an arc. We are going to prove that if X = (Ka,pba,(A,))is an inverse system in set theory of triangulated polyhedraKawith simplicial bonding functions pba and X = lim X, then there exists a uniquely determined subinverse system XX= (La, pbaLb,(A,)) of X where for each a, La is a subcomplex of Ka, each pbaLb:Lb → La is...

Yildiz, Filiz
[EN] This paper considers some various categorical aspects of the inverse systems (projective spectrums) and inverse limits described in the category ifPDitop, whose objects are ditopological plain texture spaces and morphisms are bicontinuous point functions satisfying a compatibility condition between those spaces. In this context, the category InvifPDitop consisting of the inverse systems constructed by the objects and morphisms of ifPDitop, besides the inverse systems of mappings, described between inverse systems, is introduced, and the related ideas are studied in a categorical  functorial setting. In conclusion, an identity natural transformation is obtained in the context of inverse systems ...

Alqurashi, Wafa Khalaf; Khan, Liaqat Ali; Osipov, Alexander V.
[EN] Let X and Y be topological spaces, F(X,Y) the set of all functions from X into Y and C(X,Y) the set of all continuous functions in F(X,Y). We study various setopen topologies tλ (λ ⊆ P(X)) on F(X,Y) and consider their existence, comparison and coincidence in the setting of Y a general topological space as well as for Y = R. Further, we consider the parallel notion of quasiuniform convergence topologies Uλ (λ ⊆ P(X)) on F(X,Y) to discuss Uλclosedness and right UλKcompleteness properties of a certain subspace of F(X,Y) in the case of Y a locally symmetric quasiuniform...

Ahmadullah, Md; Imdad, Mohammad; Arif, Mohammad
[EN] In this paper, we prove coincidence and common fixed points results under nonlinear contractions on a metric space equipped with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those are contained in Berzig [J. Fixed Point Theory Appl. 12, 221238 (2012))] and Alam and Imdad [To appear in Filomat (arXiv:1603.09159 (2016))]. Interestingly, a corollary to one of our main results under symmetric closure of a binary relation remains a sharpened version of a theorem due to Berzig. Finally, we use examples to highlight the accomplished improvements in the results of this paper.

Al Shumrani, M. A.
[EN] The concept of partially topological group was recently introduced in [3]. In this article, we define partially topological group action on partially topological space and we generalize some fundamental results from topological group action theory.

Protasov, Igor; Protasova, Ksenia
[EN] For every countable group G, there are 2ω distinct classes of coarselyequivalent subsets of G.

Zamani Bahabadi, Alireza
[EN] In this paper we introduce a new notion, named controlled shadowing property and we relate it to some notions in dynamical systems such as topological ergodicity, topologically mixing and specication properties. The relation between the controlled shadowing and chaos in sense of LiYorke is studied. At the end we give some examples to investigate the controlled shadowing property.