To digitize subspaces of the Euclidean nDspace, the present paper uses the Khalimsky (for short K-, if there is no danger of ambiguity) topology, K-adjacency and K-localized neighborhoods of points in Z(n), where Zn represents the set of points in the Euclidean nD space with integer coordinates. Namely, given a point p epsilon Z(n), the paper first develops a K-localized neighborhood of p epsilon Zn, denoted by NK (p) in R-n, which is substantially used in digitizing subspaces of the Euclidean nD 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-map in this paper) which need not be a continuous map under K-topology and further, develops a certain CP-isomorphism, e.g., an A-isomorphism in this paper. It turns out that an A-map overcomes some limitations of both a K-continuous map and a Khalimsky adjacency map (for brevity K A-map) so that both an A-map and an 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-map and a K-localized neighborhood, we further develop the notions of a lattice-based A-map (for short LA-map) and a lattice-based A-isomorphism (for brevity LA-isomorphism) which are used for digitizing subspaces of the Euclidean nD space in the 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.