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 .

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 Fixed Point Theory for Digital k-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1244 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Euler Characteristic ◽  
Fixed Point Theory ◽  
Geometric Realization ◽  
Point Theory ◽  
Fixed Point Property ◽  
Point Property ◽  
Euler Characteristics ◽  
Surface Theory ◽  
Closed Surfaces

The present paper studies the fixed point property (FPP) for closed k-surfaces. We also intensively study Euler characteristics of a closed k-surface and a connected sum of closed k-surfaces. Furthermore, we explore some relationships between the FPP and Euler characteristics of closed k-surfaces. After explaining how to define the Euler characteristic of a closed k-surface more precisely, we confirm a certain consistency of the Euler characteristic of a closed k-surface and a continuous analog of it. In proceeding with this work, for a simple closed k-surface in Z 3 , say S k , we can see that both the minimal 26-adjacency neighborhood of a point x ∈ S k , denoted by M k ( x ) , and the geometric realization of it in R 3 , denoted by D k ( x ) , play important roles in both digital surface theory and fixed point theory. Moreover, we prove that the simple closed 18-surfaces M S S 18 and M S S 18 ′ do not have the almost fixed point property (AFPP). Consequently, we conclude that the triviality or the non-triviality of the Euler characteristics of simple closed k-surfaces have no relationships with the FPP in digital topology. Using this fact, we correct many errors in many papers written by L. Boxer et al.

scholarly journals On \(\ell_1\)-preduals distant by 1

2018 ◽  
Vol 72 (2) ◽  
pp. 41
Author(s):  
Łukasz Piasecki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Wide Class ◽  
Fixed Point Theory ◽  
General Setting ◽  
Point Theory ◽  
Fixed Point Property ◽  
Sufficient Condition ◽  
Point Property ◽  
Weak Fixed Point Property

For every predual \(X\) of \(\ell_1\) such that the standard basis in \(\ell_1\) is weak\(^*\) convergent, we give explicit models of all Banach spaces \(Y\) for which the Banach-Mazur distance \(d(X,Y)=1\). As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space \(\ell_1\), with a predual \(X\) as above, has the stable weak\(^*\) fixed point property if and only if it has almost stable weak\(^*\) fixed point property, i.e. the dual \(Y^*\) of every Banach space \(Y\) has the weak\(^*\) fixed point property (briefly, \(\sigma(Y^*,Y)\)-FPP) whenever \(d(X,Y)=1\). Then, we construct a predual \(X\) of \(\ell_1\) for which \(\ell_1\) lacks the stable \(\sigma(\ell_1,X)\)-FPP but it has almost stable \(\sigma(\ell_1,X)\)-FPP, which in turn is a strictly stronger property than the \(\sigma(\ell_1,X)\)-FPP. Finally, in the general setting of preduals of \(\ell_1\), we give a sufficient condition for almost stable weak\(^*\) fixed point property in \(\ell_1\) and we prove that for a wide class of spaces this condition is also necessary.

Fixed Point in Minimal Spaces

2005 ◽  
Vol 10 (4) ◽  
pp. 305-314 ◽  
Author(s):  
M. Alimohammady ◽  
M. Roohi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Open Cover ◽  
Fixed Point Property ◽  
Point Property

This paper deals with fixed point theory and fixed point property in minimal spaces. We will prove that under some conditions f : (X,M) → (X,M) has a fixed point if and only if for each m-open cover {Bα} for X there is at least one x ∈ X such that both x and f(x) belong to a common Bα. Further, it is shown that if (X,M) has the fixed point property, then its minimal retract subset enjoys this property.

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.

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