Some properties of Pre-topological groups

TL;DR

This paper introduces and explores properties of pre-topological groups, extending classical theorems, and investigates their structure, embeddings, and invariants in the context of generalized topological group concepts.

Contribution

It defines pre-topological groups and generalizations, extends Markov's theorem to almost topological groups, and studies embeddings, invariants, and properties like precompactness.

Findings

Every T0 pre-topological group is regular.

Almost topological groups are completely regular.

An almost topological group is τ-narrow iff it embeds into a product of groups of weight ≤ τ.

Abstract

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.