To digitize subspaces of the Euclidean $$n$$ n D space, the present paper uses the Khalimsky (for short $$K$$ K -, if there is no danger of ambiguity) topology, $$K$$ K -adjacency and $$K$$ K -localized neighborhoods of points in $$\mathbf{Z}^n$$ Z n , where $$\mathbf{Z}^n$$ Z n represents the set of points in the Euclidean $$n$$ n D space with integer coordinates. Namely, given a point $$p \in \mathbf{Z}^n$$ p ∈ Z n , the paper first develops a $$K$$ K -localized neighborhood of $$p \in \mathbf{Z}^n$$ p ∈ Z n , denoted by $$N_K(p)$$ N K ( p ) in $$\mathbf{R}^n$$ R n , which is substantially used in digitizing subspaces of the Euclidean $$n$$ n D space. The recent paper Han and Sostak in (Comput Appl Math 32(3):521–536, 2013) proposes a connectedness preserving map (for short CP-map, e.g., an $$A$$ A -map in this paper) which need not be a continuous map under $$K$$ K -topology and further, develops a certain CP-isomorphism, e.g., an $$A$$ A -isomorphism in this paper. It turns out that an $$A$$ A -map overcomes some limitations of both a $$K$$ K -continuous map and a Khalimsky adjacency map (for brevity $$KA$$ K A -map) so that both an $$A$$ A -map and an $$A$$ A -isomorphism can substantially contribute to applied topology including both digital topology and digital geometry Han and Sostak in (Comput Appl Math 32(3):521–536, 2013). Using both an $$A$$ A -map and a $$K$$ K -localized neighborhood, we further develop the notions of a lattice-based $$A$$ A -map (for short $$LA$$ L A -map) and a lattice-based $$A$$ A -isomorphism (for brevity $$LA$$ L A -isomorphism) which are used for digitizing subspaces of the Euclidean $$n$$ n D space in the $$K$$ K -topological approach. Thus, this approach can contribute to certain branches of applied topology and computer science such as image analysis, image processing, and mathematical morphology.