Applied General Topology - Vol 10, No 1 (2009)
https://riunet.upv.es:443/handle/10251/86541
2019-07-19T00:12:20ZTopologies on function spaces and hyperspaces
https://riunet.upv.es:443/handle/10251/86556
Topologies on function spaces and hyperspaces
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.
2017-09-06T12:06:42ZWell-posedness, bornologies, and the structure of metric spaces
https://riunet.upv.es:443/handle/10251/86555
Well-posedness, bornologies, and the structure of metric spaces
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 UC-spaces 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 bornologies that can be expressed as Bλ for some λ, as well as those that can be so induced by a uniformly continuous λ.
2017-09-06T11:59:49ZNew coincidence and common fixed point theorems
https://riunet.upv.es:443/handle/10251/86553
New coincidence and common fixed point theorems
Singh, S.L.; Hematulin, Apichai; Pant, Rajendra
[EN] In this paper, we obtain some extensions and a generalization of a remarkable fixed point theorem of Proinov. Indeed, we obtain some coincidence and fixed point theorems for asymptotically regular non-self and self-maps without requiring continuity and relaxing the completeness of the space. Some useful examples and discussions are also given.
2017-09-06T11:54:20ZArnautov's problems on semitopological isomorphisms
https://riunet.upv.es:443/handle/10251/86552
Arnautov's problems on semitopological isomorphisms
Dikranjan, Dikran; Giordano Bruno, Anna
[EN] Semitopological isomorphisms of topological groups were introduced by Arnautov [2], who posed several questions related to compositions of semitopological isomorphisms and about the groups G (we call them Arnautov groups) such that for every group topology on G every semitopological isomorphism with domain (G, ) is necessarily open (i.e., a topological isomorphism). We propose a different approach to these problems by introducing appropriate new notions, necessary for a deeper understanding of Arnautov groups. This allows us to find some partial answers and many examples. In particular, we discuss the relation with minimal groups and non-topologizable groups.
2017-09-06T11:51:42ZF-supercontinuous functions
https://riunet.upv.es:443/handle/10251/86551
F-supercontinuous functions
Kohli, J.K.; Singh, D.; Aggarwal, Jeetendra
[EN] A strong variant of continuity called ‘F-supercontinuity’ is introduced. The class of F-supercontinuous functions strictly contains the class of z-supercontinuous functions (Indian J. Pure Appl. Math. 33 (7) (2002), 1097–1108) which in turn properly contains the class of cl-supercontinuous 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 F-supercontinuous functions is properly contained in the class of R-supercontinuous functions which in turn is strictly contained in the class of continuous functions. Basic properties of F-supercontinuous functions are studied and their place in the hierarchy of strong variants of continuity, which already exist in the mathematical literature, is elaborated. If either domain or range is a functionally regular space (Indagationes Math. 15 (1951), 359–368; 38 (1976), 281–288), then the notions of continuity, F-supercontinuity and R-supercontinuity coincide.
2017-09-06T11:49:12ZPointwise convergence and Ascoli theorems for nearness spaces
https://riunet.upv.es:443/handle/10251/86550
Pointwise convergence and Ascoli theorems for nearness spaces
Yang, Zhanbo
[EN] We first study subspaces and product spaces in the context of nearness spaces and prove that U-N spaces, C-N spaces, PN spaces and totally bounded nearness spaces are nearness hereditary; T-N spaces and compact nearness spaces are N-closed hereditary. We prove that N2 plus compact implies N-closed 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 sub-categories. We prove that a net of functions is convergent under the pointwise convergent nearness structure if and only if its cross-section at each point is convergent. We have also proved two Ascoli-Arzelà type of theorems.
2017-09-06T11:46:45ZCondensations of Cp(X) onto σ-compact spaces
https://riunet.upv.es:443/handle/10251/86548
Condensations of Cp(X) onto σ-compact spaces
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 Cp-theory. We prove, among other things, that if K is a non-metrizable Corson compact space then Cp(K) cannot be condensed onto a σ-compact space. This answers two questions published by Arhangel’skii and Pavlov.
2017-09-06T11:44:00Z∗-half completeness in quasi-uniform spaces
https://riunet.upv.es:443/handle/10251/86547
∗-half completeness in quasi-uniform spaces
Andrikopoulos, Athanasios
[EN] Romaguera and Sánchez-Granero (2003) have introduced the notion of T1∗-half completion and used it to see when a quasi-uniform space has a ∗-compactification. In this paper, for any quasi-uniform space, we construct a ∗-half completion, called standard ∗-half completion. The constructed ∗-half completion coincides with the usual uniform completion in the uniform spaces and is the unique (up to quasi-isomorphism) T1 ∗-half completion of a symmetrizable quasi-uniform space. Moreover, it constitutes a ∗-compactification for ∗-Cauchy bounded quasi-uniform spaces. Finally, we give an example which shows that the standard ∗-half completion differs from the bicompletion construction.
2017-09-06T11:40:52ZBest proximity pair theorems for relatively nonexpansive mappings
https://riunet.upv.es:443/handle/10251/86546
Best proximity pair theorems for relatively nonexpansive mappings
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). Using Zorn’s lemma, Eldred et.al proved that such a point x0 exists in a uniformly convex Banach space settings under the conditions stated above. In this paper, by using a convergence theorem we attempt to prove the existence of such a point x0 (called best proximity point) without invoking Zorn’s lemma.
2017-09-06T11:37:36ZEmbedding into discretely absolutely star-Lindelöf spaces II
https://riunet.upv.es:443/handle/10251/86545
Embedding into discretely absolutely star-Lindelöf spaces II
Song, Yan-Kui
[EN] A space X is discretely absolutely star-Lindelöf 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 star-Lindelöf space can be represented in a Hausdorff discretely absolutely star-Lindelöf space as a closed G-subspace.
2017-09-06T11:35:00Z