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Abstract. Let X n,k be a Khalimsky topological n dimensional subspace with digital k-connectivity. In relation to the classification of spaces X n,k , by comparing several kinds of continuities and homeomorphisms, the paper proposes a category which is suitable for studying the spaces X n,k .

Abstract. Several kinds of homotopies have been substantially used to study topological properties of digital spaces. The present paper, as a survey article, studies some recent results in the field of homotopy theory associated with Khalimsky topology. In particular, Khalimsky topological properties of digital products related to the establishment of the homotopies are mainly treated.