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

Abstract

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.