  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.
