Links between Contractibility and Fixed Point Property for Khalimsky Topological Spaces

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 .