Covering Maps with respect to Topologies on the Fundamental Group

In this paper, using the classical covering theory, we introduce a generalization of covering maps of a space $X$ with respect to a topology $\tau$ on the fundamental group of $X$. We show that the famous notions, covering, semicovering, generalized covering and fibration maps are of special cases of this new notion ${\pi }_1^{\tau }$-covering map. Moreover, among presenting some properties for this new notion, we compare ${\pi }_1^{\tau }$-covering maps of a space $X$ for several famous topologies on the fundamental group of $X$.