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, κ n X) be an n-dimensional Khalimsky topological space induced by the n-dimensional Khalimsky space denoted by (Z n , κ n). Although not every connected Khalimsky topological space (X, κ n X) has the FPP, we prove that for every point x ∈ Z n the smallest open K-topological neighborhood of x, denoted by SN K (x) ⊂ (Z n , κ n), 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, κ n X) := X are assumed to be connected and | X | ≥ 2.

Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations

“…The author in [8,10] proved the FPP of the smallest open neighborhood of (Z n , κ n ) [10] and the non-FPP of a compact M-topological plane in (Z 2 , γ) [8]. Thus, we may now pose the following queries about the AFPP of compact M-topological plane X and the preservation of the AFPP of a compact n-dimensional Euclidean space (or cube) into that of each of K-, M-, Uand L-digitization, as follows:…”

“…Regarding Questions 1 and 3, the author in [10] proved the FPP of SN K (p) in (Z n , κ n ). Moreover, the authors in [13] proved that the functor D K preserves the connectedness of (X, κ n X ) into its K-digitized space (D K (X), κ n D K (X) ).…”

“…We say that a digital space is a nonempty, π-connected, symmetric relation set, denoted by (X, π) [11]. It is well known that a digital space [11] includes a digital image (X, k) with digital k-connectivity (i.e., Rosenfeld model) [2,14], a Khalimsky (K-, for brevity) topological space with Khalimsky adjacency [15], a Marcus-Wyse (M-, for short) topological space with Marcus-Wyse adjacency [16], and so forth [5,9,10] (see Section 2 in details).…”

“…For a point x in (X, γ X ), we denote by SN M (x) the smallest open neighborhood of x in (X, γ X ). For (X, γ X ), we say that distinct points x and y in X are [10], where SN M (p) is the smallest open set containing the point p in (X, γ X ). According to this M-adjacency, it turns out that an M-topological space (X, γ X ) is a digital space [9].…”

“…Using the set of M-topological spaces (X, γ X ) and that of M-continuous maps for every ordered pair of objects of M-topological spaces, we obtain the category of M-topological spaces, denoted by MTC [10]. Remark 1.…”