Some properties of pre-topological groups

In this paper, the concepts of pre-topological groups and some generalizations of pre- topological groups are posed. First, some basic properties of pre-topological groups are systematically investigated; in particular, we prove that each T 0 pre-topological group is regular, and every T 0 almost topological group is completely regular which extends A.A. Markov's theorem to the class of almost topological groups. Moreover, it is shown that an almost topological group is tau- narrow if and only if it can be embedded as a subgroup of a pre-topological C- pro duct of almost topological groups of weight less than or equal to tau. Finally, the cardinal invariant, the precompactness and the resolvability are investigated in the class of pre-topological groups. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.