Existence of a proper subspace of [formula omitted] which is homeomorphic to the n-dimensional Khalimsky topological space

Abstract

Is there a topology on the set of integers whose proper subtopology is homeomorphic to the Khalimsky line topology? To address this open problem, the present paper focuses on the topologies TSk and TSk′ on the set of integers, k∈Z. The former is generated by the set Sk:={Sk,t|Sk,t:={2t,2t+1,2t+2k+1},t∈Z} and the latter is generated by the set Sk′:={Sk,t′|Sk,t′:={2t,2t+1,2t+2k},t∈Z} as a subbase. Then, we first investigate some properties of semi-open and semi-closed subsets of these two topological spaces (Z,TSk) and (Z,TSk′). Next, we prove that for k∈Z∖{0} the topological space (Z,TSk) (resp. (Z,TSk′)) has |k| numbers of proper subspaces Pi:=(Pi,(TSk)Pi) (resp. Ci:=(Ci,(TSk′)Ci)), i∈[1,k]Z, such that each of them is homeomorphic to the Khalimsky line topological space. Besides, for i∈[1,k]Z, each of all Pi and Ci is proved to be homeomorphic to the other. Finally, we prove that for k∈N each of the product topological spaces (Zn,(TSk)n) and (Zn,(TSk′)n) has a proper subspace which is homeomorphic to the n-dimensional Khalimsky topological space.