Applied General Topology

Selection principles and bitopological hyperspaces

2023-02-06T10:30:13+01:00

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^X the family of all closed subsets of X. We investigate selection properties of the bitopological space (2^X, Δ₁⁺ , Δ₂⁺) where Δ_i⁺ is the upper Δ_i-topology for each i=1,2.

Fixed points which belong to the set of unit values of a suitable function on fuzzy metric spaces

2022-09-16T09:49:43+02:00

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.

Good coverings of proximal Alexandrov spaces. Path cycles in the extension of the Mitsuishi-Yamaguchi good covering and Jordan Curve Theorems

2022-09-29T22:55:20+02:00

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 cycle is a sequence of maps h₁,...,h_i,...,h_n-1 mod n in which h_i : [ 0,1 ] → X and h_i(1) = h_i+1(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.

Digital semicovering and digital quasicovering maps

2022-08-18T01:06:26+02:00

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.

Uniformly refinable maps

2022-09-29T22:51:27+02:00

We introduce the notion of uniformly refinable map for compact, Hausdorff spaces, as a generalization of refinable maps originally
defined for metric continua by Jo Ford (Heath) and Jack W. Rogers, Jr., Refinable maps, Colloq. Math., 39 (1978), 263-269.

C(X) determines X - an inherent theory

2022-07-18T19:23:14+02:00

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.

Proper spaces are spectral

2022-11-30T15:09:43+01:00

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.

Best proximity point for q-ordered proximal contraction in noncommutative Banach spaces

2022-08-29T14:30:42+02:00

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.

On setwise betweenness

2022-07-19T19:33:38+02:00

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.

A counter example on a Borsuk conjecture

2022-08-08T17:02:45+02:00

The study of shape restrictions of subsets of R^d 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.

Common fixed point results for a generalized ( ψ, φ )-rational contraction

2022-11-18T22:17:41+01:00

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.

Interpolative contractions and discontinuity at fixed point

2022-11-14T19:12:41+01:00

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)]^β[d(x,Ty)]^α[d(x,Ty)]^γ[(d(x,Ty)+d(x,Ty))/2]^{1−α−β−γ}, where α , β , γ ∈ ( 0,1 ) with α + β + γ < 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.

Hybrid topologies on the real line

2022-11-15T09:27:46+01:00

Given A ⊆ ℝ , the Hattori space H(A) is the topological space ( ℝ , τ_A ) where each a ∈ A has a τ_A -neighborhood base { ( a − ε , a + ε ) : ε > 0 } and each b ∈ ℝ − A has a τ_A -neighborhood base { [ b , b + ε ) : ε > 0 } . Thus, τ_A 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.

New results regarding the lattice of uniform topologies on C(X)

2022-11-09T23:07:03+01:00

For a Tychonoff space X, the lattice U_X was introduced in L. A. Pérez-Morales, G. Delgadillo-Piñón, and R. Pichardo-Mendoza, The lattice of uniform topologies on C(X), Questions and Answers in General Topology 39 (2021), 65-71.

In the present paper we continue the study of U_X. To be specific, the present paper deals, in its first half, with structural and categorical properties of U_X, while in its second part focuses on cardinal characteristics of the lattice and how these relate to some cardinal functions of the space X.

Concrete functors that respect initiality and finality

2022-11-16T12:25:40+01:00

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.

Τ-quasi-Cauchy spaces - a non-symmetric theory of completeness and completion

2023-02-08T19:26:09+01:00

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.

Some network-type properties of the space of G-permutation degree

2022-12-21T07:28:14+01:00

In this paper the network-type properties (network,cs−network,cs∗−network,cn−network andck−network) of the spaceSP_nGXofG-permutation degree ofXare studied. It is proved that:(1) IfXis aT₁-space that has a network of cardinality≤κ, thenSP_nGXhas a network of cardinality≤κ;(2) IfXis aT₁-space that has acs-network (resp.cs∗-network) ofcardinality≤κ, thenSP_nGXhas acs-network (resp.cs∗-network) ofcardinality≤κ;(3) IfXis aT₁-space that has acn-network (resp.ck-network) ofcardinality≤κ, thenSP_nGXhas acn-network (resp.ck−network) ofcardinality≤κ.