Some properties of Pre-topological groups

In this paper, we pose the concepts of pre-topological groups and some generalizations of pre-topological groups. First, we systematically investigate some basic properties of pre-topological groups; in particular, we prove that each $T_{0}$ pre-topological group is regular and every 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 product 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.