scholarly journals The Fixed Point Property of the Infinite M-Sphere

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 599
Author(s):  
Sang-Eon Han ◽  
Selma Özçağ
Keyword(s):  
Applied Sciences ◽  
Fixed Point ◽  
Topological Space ◽  
Unsolved Problem ◽  
Fixed Point Property ◽  
Digital Geometry ◽  
Point Property ◽  
Open Set ◽  
One Point Compactification ◽  
Digital Or

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space ( Z 2 , γ ) . This compactification is called the infinite M-topological sphere and denoted by ( ( Z 2 ) ∗ , γ ∗ ) , where ( Z 2 ) ∗ : = Z 2 ∪ { ∗ } , ∗ ∉ Z 2 and γ ∗ is the topology for ( Z 2 ) ∗ induced by the topology γ on Z 2 . With the topological space ( ( Z 2 ) ∗ , γ ∗ ) , since any open set containing the point “ ∗ ” has the cardinality ℵ 0 , we call ( ( Z 2 ) ∗ , γ ∗ ) the infinite M-topological sphere. Indeed, in the fields of digital or computational topology or applied analysis, there is an unsolved problem as follows: Under what category does ( ( Z 2 ) ∗ , γ ∗ ) have the fixed point property (FPP, for short)? The present paper proves that ( ( Z 2 ) ∗ , γ ∗ ) has the FPP in the category M o p ( γ ∗ ) whose object is the only ( ( Z 2 ) ∗ , γ ∗ ) and morphisms are all continuous self-maps g of ( ( Z 2 ) ∗ , γ ∗ ) such that | g ( ( Z 2 ) ∗ ) | = ℵ 0 with ∗ ∈ g ( ( Z 2 ) ∗ ) or g ( ( Z 2 ) ∗ ) is a singleton. Since ( ( Z 2 ) ∗ , γ ∗ ) can be a model for a digital sphere derived from the M-topological space ( Z 2 , γ ) , it can play a crucial role in topology, digital geometry and applied sciences.

scholarly journals A Counterexample in Finite Fixed Point Theory

1979 ◽  
Vol 22 (1) ◽  
pp. 99-100
Author(s):  
H. C. Enos
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Property ◽  
Point Property ◽  
Closed Sets

This note answers a question raised by Lee Mohler in 1970, by exhibiting a finite topological space X which is the union of closed subspaces Y, Z, such that Y, Z, and Y ⋂ Z, but not X, have the fixed point property. The example is a triangulation △ of S3, the points of X being the simplices of Δ and the closed sets the subcomplexes of △.

scholarly journals On Almost-Fixed-Point Theory

1978 ◽  
Vol 30 (4) ◽  
pp. 673-699 ◽  
Author(s):  
Michiel Hazewinkel ◽  
Marcel Van De Vel
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Property ◽  
Point Property ◽  
Finite Covering ◽  
Continuous Maps

Let X be a topological space, a finite covering of X (the words ‘covering’ and ‘cover’ are used interchangeably). We say that has the almost fixed point property for a class of continuous maps f : X → X if for all there is an x ∈ X and such that x ∈ U and f(x) ∈ U, or, equivalently, if there is a such that .

scholarly journals The fixed point property of the infinite K-sphere in the set Con*((Z2)*)

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4027-4042
Author(s):  
Sang-Eon Hana
Keyword(s):  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property ◽  
Topological Plane ◽  
The One ◽  
One Point Compactification

In this paper the Alexandroff one point compactification of the 2-dimensional Khalimsky (K-, for brevity) plane (resp. the 1-dimensional Khalimsky line) is called the infinite K-sphere (resp. the infinite K-circle). The present paper initially proves that the infinite K-circle has the fixed point property (FPP, for short) in the set Con(Z*), where Con(Z*) means the set of all continuous self-maps f of the infinite K-circle. Next, we address the following query which remains open: Under what condition does the infinite K-sphere have the FPP? Regarding this issue, we prove that the infinite K-sphere has the FPP in the set Con*((Z2)*) (see Definition 1.1). Finally, we compare the FPP of the infinite K-sphere and that of the infinite M-sphere, where the infinite M-sphere means the one point compactification of the Marcus-Wyse topological plane.

scholarly journals The Fixed Point Property of Non-Retractable Topological Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 879 ◽  
Author(s):  
Jeong Kang ◽  
Sang-Eon Han ◽  
Sik Lee
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Property ◽  
Topological Spaces ◽  
Point Property ◽  
Finite Planes ◽  
Topological Plane ◽  
Theoretic Foundations

Unlike the study of the fixed point property (FPP, for brevity) of retractable topological spaces, the research of the FPP of non-retractable topological spaces remains. The present paper deals with the issue. Based on order-theoretic foundations and fixed point theory for Khalimsky (K-, for short) topological spaces, the present paper studies the product property of the FPP for K-topological spaces. Furthermore, the paper investigates the FPP of various types of connected K-topological spaces such as non-K-retractable spaces and some points deleted K-topological (finite) planes, and so on. To be specific, after proving that not every one point deleted subspace of a finite K-topological plane X is a K-retract of X, we study the FPP of a non-retractable topological space Y, such as one point deleted space Y ∖ { p } .

scholarly journals Links between Contractibility and Fixed Point Property for Khalimsky Topological Spaces

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 18
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Property ◽  
Topological Spaces ◽  
Point Property ◽  
Continuous Maps

Given a Khalimsky (for short, K-) topological space X, the present paper examines if there are some relationships between the contractibility of X and the existence of the fixed point property of X. Based on a K-homotopy for K-topological spaces, we firstly prove that a K-homeomorphism preserves a K-homotopy between two K-continuous maps. Thus, we obtain that a K-homeomorphism preserves K-contractibility. Besides, the present paper proves that every simple closed K-curve in the n-dimensional K-topological space, S C K n , l , n ≥ 2 , l ≥ 4 , is not K-contractible. This feature plays an important role in fixed point theory for K-topological spaces. In addition, given a K-topological space X, after developing the notion of K-contractibility relative to each singleton { x } ( ⊂ X ) , we firstly compare it with the concept of K-contractibility of X. Finally, we prove that the K-contractibility does not imply the K-contractibility relative to each singleton { x 0 } ( ⊂ X ) . Furthermore, we deal with certain conjectures involving the (almost) fixed point property in the categories KTC and KAC, where KTC (see Section 3) (resp. KAC (see Section 5)) denotes the category of K-topological (resp. KA-) spaces, KA-) spaces are subgraphs of the connectedness graphs of the K-topology on Z n .

The fixed point property of the smallest open neighborhood of the n-dimensional Khalimsky topological space

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6165-6173 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Open Neighborhood ◽  
Fixed Point Property ◽  
Point Property

The paper aims to propose the fixed point property(FPP for short) of smallest open neighborhoods of the n-dimensional Khalimsky space and further, the FPP of a Khalimsky (K-, for short) retract. Let (X, knX) be an n-dimensional Khalimsky topological space induced by the n-dimensional Khalimsky space denoted by (Zn, kn). Although not every connected Khalimsky topological space (X, knX) has the FPP, we prove that for every point x 2 Zn the smallest open K-topological neighborhood of x, denoted by SNK(x) ? (Zn,kn), has the FPP. Besides, the present paper also studies the almost fixed point property (AFPP, for brevity) of a K-topological space. In this paper all spaces (X,knX):= X are assumed to be connected and ?X? ? 2.

scholarly journals Brush spaces and the fixed point property

2011 ◽  
Vol 158 (8) ◽  
pp. 1085-1089 ◽  
Author(s):  
M.M. Marsh ◽  
J.R. Prajs
Keyword(s):  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property

scholarly journals Towards the fixed point property for superreflexive spaces

2001 ◽  
Vol 64 (3) ◽  
pp. 435-444 ◽  
Author(s):  
Andrzej Wiśnicki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unit Sphere ◽  
Convex Set ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometrical Condition

A Banach space X is said to have property (Sm) if every metrically convex set A ⊂ X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.

scholarly journals A product space with the fixed point property

2012 ◽  
Vol 2012 (1) ◽  
Author(s):  
Helga Fetter Nathansky ◽  
Enrique Llorens-Fuster
Keyword(s):  
Fixed Point ◽  
Product Space ◽  
Fixed Point Property ◽  
Point Property
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