Applied General Topology <p style="margin-top: 0cm; margin-right: 0cm; margin-bottom: 6.0pt; margin-left: 0cm; text-align: justify; text-justify: inter-ideograph;"><span>The international journal <strong>Applied General Topology</strong> publishes only original research papers related to the interactions between General Topology and other mathematical disciplines as well as topological results with applications to other areas of Science, and the development of topological theories of sufficiently general relevance to allow for future applications.</span></p> Universitat Politècnica de València en-US Applied General Topology 1576-9402 <p><a href="" rel="license"><img src="" alt="Creative Commons License" /></a></p> <p>This journal is licensed under a <a href="" rel="license">Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License</a>.</p> Selection principles and bitopological hyperspaces <p>In this paper we continue to research relationships between closure-type properties of hyperspaces over a space X and covering properties of X. For a Hausdorff space X we denote by 2<sup>X</sup> the family of all closed subsets of X. We investigate selection properties of the bitopological space (2<sup>X</sup>, Δ<sub>1</sub><sup>+</sup> , Δ<sub>2</sub><sup>+</sup>) where Δ<sub>i</sub><sup>+</sup> is the upper Δ<sub>i</sub>-topology for each i=1,2.</p> Alexander V. Osipov Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 1 8 10.4995/agt.2023.12424 Fixed points which belong to the set of unit values of a suitable function on fuzzy metric spaces <p>In this paper, we introduce the notion of fuzzy (F, φ,β-ψ)-contractive mappings in fuzzy metric spaces and utilize the same to prove some existence and uniqueness fuzzy φ-fixed point results in both M-complete and G-complete fuzzy metric spaces. The obtained results extend, generalize and improve some relevant results of the existing literature. An illustrative example is utilized to demonstrate the usefulness and effectiveness of our results.</p> Hayel N. Saleh Mohammad Imdad Wutiphol Sintunavarat Copyright (c) 2022 Applied General Topology 2023-04-05 2023-04-05 24 1 9 24 10.4995/agt.2023.16924 Good coverings of proximal Alexandrov spaces. Path cycles in the extension of the Mitsuishi-Yamaguchi good covering and Jordan Curve Theorems <p>This paper introduces proximal path cycles, which lead to the main results in this paper, namely, extensions of the Mitsuishi-Yamaguchi Good Coverning Theorem with different forms of Tanaka good cover of an Alexandrov space equipped with a proximity relation as well as extension of the Jordan curve theorem. In this work, a path <strong>cycle</strong> is a sequence of maps h<sub>1</sub>,...,h<sub>i</sub>,...,h<sub>n-1</sub> <em>mod n</em> in which h<sub>i</sub> : [ 0,1 ] → X and h<sub>i</sub>(1) = h<sub>i+1</sub>(0) provide the structure of a path-connected cycle that has no end path. An application of these results is also given for the persistence of proximal video frame shapes that appear in path cycles.</p> James Francis Peters Tane Vergili Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 25 45 10.4995/agt.2023.17046 Digital semicovering and digital quasicovering maps <p>In this paper we introduce notions of digital semicovering and digital quasicovering maps. We show that these are generalizations of digital covering maps and investigate their relations. We will also clarify the relationship between these generalizations and digital path lifting.</p> Ali Pakdaman Copyright (c) 2022 Applied General Topology 2023-04-05 2023-04-05 24 1 47 57 10.4995/agt.2023.17156 Uniformly refinable maps <p>We introduce the notion of uniformly refinable map for compact, Hausdorff spaces, as a generalization of refinable maps originally<br />defined for metric continua by Jo Ford (Heath) and Jack W. Rogers, Jr., Refinable maps, Colloq. Math., 39 (1978), 263-269.</p> Sergio Macías Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 59 81 10.4995/agt.2023.17345 C(X) determines X - an inherent theory <p>One of the fundamental problem in rings of continuous function is to extract those spaces for which C(X) determines X, that is to investigate X and Y such that C(X) isomorphic with C(Y ) implies X homeomorphic with Y. The development started back from Tychonoff who first pointed out inevitability of Tychonoff space in this category of problem. Later S. Banach and M. Stone proved independently with slight variance, that if X is compact Hausdorff space, C(X) also determine X. Their works were maximally extended by E. Hewitt by introducing realcompact spaces and later Melvin Henriksen and Biswajit Mitra solved the problem for locally compact and nearly realcompact spaces. In this paper we tried to develop an inherent theory of this problem to cover up all the works in the literature introducing a notion so called P-compact spaces.</p> Biswajit Mitra Sanjib Das Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 83 93 10.4995/agt.2023.17569 Proper spaces are spectral <p>Since Hochster's work, spectral spaces have attracted increasing interest. Through this note we give a new self-contained and constructible topology-independent proof of the fact that the set of proper ideals of a ring endowed with coarse lower topology is a spectral space.</p> Amartya Goswami Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 95 99 10.4995/agt.2023.17800 Best proximity point for q-ordered proximal contraction in noncommutative Banach spaces <p>We introduce the concept of q-ordered proximal nonunique contraction for the non self mappings and then obtain some proximity point results for these mappings. We also furnish examples to support our claims.</p> Ayush Bartwal Shivam Rawat Ismat Beg Copyright (c) 2022 Applied General Topology 2023-04-05 2023-04-05 24 1 101 113 10.4995/agt.2023.18029 On setwise betweenness <p>In this article, we investigate the notion of setwise betweenness, a concept introduced by P. Bankston as a generalisation of pointwise betweenness. In the context of continua, we say that a subset C of a continuum X is between distinct points a and b of X if every subcontinuum K of X containing both a and b intersects C. The notion of an interval [a,b] then arises naturally. Further interesting questions are derived from considering such intervals within an associated hyperspace on X. We explore these ideas within the context of the Vietoris topology and n-symmetric product hyperspaces on all nonempty closed subsets of a topological space X, CL(X). Moreover, an alternative pointwise interval, derived from setwise intervals, is introduced.</p> Qays R. Shakir Copyright (c) 2022 Applied General Topology 2023-04-05 2023-04-05 24 1 115 123 10.4995/agt.2023.18061 A counter example on a Borsuk conjecture <p>The study of shape restrictions of subsets of R<sup>d</sup> has several applications in many areas, being convexity, r-convexity, and positive reach, some of the most famous, and typically imposed in set estimation. The following problem was attributed to K. Borsuk, by J. Perkal in 1956:find an r-convex set which is not locally contractible. Stated in that way is trivial to find such a set. However, if we ask the set to be equal to the closure of its interior (a condition fulfilled for instance if the set is the support of a probability distribution absolutely continuous with respect to the d-dimensional Lebesgue measure), the problem is much more difficult. We present a counter example of a not locally contractible set, which is r-convex. This also proves that the class of supports with positive reach of absolutely continuous distributions includes strictly the class ofr-convex supports of absolutely continuous distributions.</p> Alejandro Cholaquidis Copyright (c) 2022 Applied General Topology 2023-04-05 2023-04-05 24 1 125 128 10.4995/agt.2023.18176 Common fixed point results for a generalized ( ψ, φ )-rational contraction <p>In this paper, we obtain two common fixed point results for mappings satisfying the generalized (ψ,φ)-contractive type conditions given by a rational expression on a complete metric space. Our results generalize several well known theorems of the literature in the context of (ψ,φ)-rational contraction. In addition, there is an example for obtained results.</p> M. C. Arya N. Chandra Mahesh C. Joshi Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 129 144 10.4995/agt.2023.18320 Interpolative contractions and discontinuity at fixed point <p>In this paper, we investigate new solutions to the Rhoades' discontinuity problem on the existence of a self-mapping which has a fixed point but is not continuous at the fixed point on metric spaces. To do this, we use the number defined as n(x,y)=[d(x,y)]<sup>β</sup>[d(x,Ty)]<sup>α</sup>[d(x,Ty)]<sup>γ</sup>[(d(x,Ty)+d(x,Ty))/2]<sup>1−α−β−γ</sup>, where α , β , γ ∈ ( 0,1 ) with α + β + γ &lt; 1 and some interpolative type contractive conditions. Also, we investigate some geometric properties of Fix(T) under some interpolative type contractions and prove some fixed-disc (resp. fixed-circle) results. Finally, we present a new application to the discontinuous activation functions.</p> Nihal Taş Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 145 156 10.4995/agt.2023.18552 Hybrid topologies on the real line <p>Given A ⊆ ℝ , the Hattori space H(A) is the topological space ( ℝ , τ<sub>A</sub> ) where each a ∈ A has a τ<sub>A</sub> -neighborhood base { ( a − ε , a + ε ) : ε &gt; 0 } and each b ∈ ℝ − A has a τ<sub>A</sub> -neighborhood base { [ b , b + ε ) : ε &gt; 0 } . Thus, τ<sub>A</sub> may be viewed as a hybrid of the Euclidean topology and the lower-limit topology. We investigate properties of Hattori spaces as well as other hybrid topologies on ℝ using various combinations of the discrete, left-ray, lower-limit, upper-limit, and Euclidean topologies. Since each of these topologies is generated by a quasi-metric on ℝ, we investigate hybrid quasi-metrics which generate these hybrid topologies.</p> Tom Richmond Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 157 168 10.4995/agt.2023.18566 New results regarding the lattice of uniform topologies on C(X) <p>For a Tychonoff space X, the lattice U<sub>X</sub> was introduced in L. A. Pérez-Morales, G. Delgadillo-Piñón, and R. Pichardo-Mendoza, <em>The lattice of uniform topologies on </em><em>C(X)</em>, Questions and Answers in General Topology 39 (2021), 65-71.</p> <p>In the present paper we continue the study of U<sub>X</sub>. To be specific, the present paper deals, in its first half, with structural and categorical properties of U<sub>X</sub>, while in its second part focuses on cardinal characteristics of the lattice and how these relate to some cardinal functions of the space X.</p> Roberto Pichardo-Mendoza Alejandro Ríos-Herrejón Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 169 185 10.4995/agt.2023.18738 Concrete functors that respect initiality and finality <div id="magicparlabel-63800" class="abstract"> <div class="abstract_item">We study concrete endofunctors of the category of convergence spaces and continuous maps that send initial maps to initial maps or final maps to final maps. The former phenomenon turns out to be fairly common while the latter is rare. In particular, it is shown that the pretopological modification is the coarsest hereditary modifier finer than the topological modifier and this is applied to give a structural interpretation of the role of Fréchet-Urysohn spaces with respect to sequential spaces and of k' -spaces with respect to k -spaces.</div> </div> Frédéric Mynard Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 187 203 10.4995/agt.2023.18771 Τ-quasi-Cauchy spaces - a non-symmetric theory of completeness and completion <p>Based on the concept of Cauchy pair Τ-filters, we develop an axiomatic theory of completeness for non-symmetric spaces, such as Τ-quasi-uniform (limit) spaces or L-metric spaces. We show that the category of Τ-quasi-Cauchy spaces is topological and Cartesian closed and we construct a finest completion for a non-complete Τ-quasi-Cauchy space. In the special case of symmetry, Τ-quasi-Cauchy spaces can be identified with Τ-Cauchy spaces.</p> Gunther Jäger Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 205 227 10.4995/agt.2023.18783 Some network-type properties of the space of G-permutation degree <p>In this paper the network-type properties (network,cs−network,cs∗−network,cn−network andck−network) of the spaceSP<sub>nG</sub>XofG-permutation degree ofXare studied. It is proved that:(1) IfXis aT<sub>1</sub>-space that has a network of cardinality≤κ, thenSP<sub>nG</sub>Xhas a network of cardinality≤κ;(2) IfXis aT<sub>1</sub>-space that has acs-network (resp.cs∗-network) ofcardinality≤κ, thenSP<sub>nG</sub>Xhas acs-network (resp.cs∗-network) ofcardinality≤κ;(3) IfXis aT<sub>1</sub>-space that has acn-network ( ofcardinality≤κ, thenSP<sub>nG</sub>Xhas acn-network (−network) ofcardinality≤κ.</p> Lj. D.R Kočinac F. G. Mukhamadiev A. K. Sadullaev Sh. U. Meyliev Copyright (c) 2023 Applied General Topology 2023-04-05 2023-04-05 24 1 229 237 10.4995/agt.2023.18985