Links between Contractibility and Fixed Point Property for Khalimsky Topological Spaces

Abstract

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 infinite K-sphere in the set Con*((Z2)*)

“…Naively, we may ask if for any f ∈ Con(Z * ) there is some point x ∈ Z * such that f (x) = x. Besides, a K-topological invariant [8] is said to be a property of a K-topological space that is invariant under K-homeomorphisms. In other words, we often call that property is a K-topological property [8].…”

“…Besides, a K-topological invariant [8] is said to be a property of a K-topological space that is invariant under K-homeomorphisms. In other words, we often call that property is a K-topological property [8]. This section is devoted to proving the FPP of (Z * , κ * ) in the set Con(Z * ) (see Theorem 4.3).…”

“…Hence (Im( f ), κ Im( f ) ) should be a finite (simple) K-path on (Z, κ) which is a subspace of (Z, κ). Since any finite K-path in (Z, κ) has the FPP [8] and further, the FPP is the K-topological property [8], (Im( f ), κ Im( f ) ) = (Im( f ), κ * Im( f ) ) has the FPP with respect to the map f . By Lemma 3.8 and Proposition 3.9, we conclude that the given map f has a point x ∈ Z ⊂ Z * such that f (x) = x.…”

“…Regarding this issue, up to now, only a few cases were proved. For instance, in the case Im( f ) is one of the smallest open sets [6] or it is equal to a K-retractable subspace of the K-square (I 2 , κ 2 I 2 ) [13], it turns out that (Im( f ), κ 2 Im( f ) ) has the FPP [8,13]. Thus, we need to investigate a certain subset of Con((Z 2 ) * ) supporting the FPP for (Im( f ), (κ 2 ) * Im( f ) ) or the infinite K-sphere.…”