Strong homotopy in finite topological adjacency category

The present paper investigates a strong homotopy (i.e., SA-homotopy) in a finite topological adjacency category. We prove that two minimal finite spaces are SA-homotopy equivalent if and only if they are A-isomorphic in the category, and that two finite T o -spaces are SA-homotopy equivalent if and only if they have A-isomorphic cores. As an application, we prove that all simple closed KA-curves are not SA-contractible. The SA-homotopy is a generalization of the classical topological homotopy. We also reveal similar properties between SA-homotopy and the classical topological homotopy. Moreover, in the sense of SA-homotopy we answer two questions having close relationships with that posed by Boxer (2005) [4]. (C) 2021 Elsevier B.V. All rights reserved.