if we consider some special conditions, we can assume fundamental group of a topological space as a new topological space. in this paper, we will present a number of theorems in topological fundamental group related to semilocally simply connected property for a topological space.

If we consider some special conditions, we can assume fundamental group of a topological space as a new topological space. In this paper, we will present a number of theorems in topological fundamental group related to semilocally simply connected property for a topological space.

In this paper, we introduce the based and unbased n-Spanier groups n-semilocally simply connected spaces, investigate their relationship. We show that under some conditions, vanishing of group with respect to an open cover is equivalent n-semilocal simple connectivity space, vice versa.

The concept of a topology for the fundamental group appears to have originated with Hurewicz (1935), but it has been actively studied only for the last ten years. The classical fundamental group of a pointed topological space X naturally inherits the quotient topology from the space of based loops in X endowed with the compact-open topology. The resulting space, called the topological fundament...

For a locally path connected topological space, the topological fundamental group is discrete if and only if the space is semilocally simplyconnected. While functoriality of the topological fundamental group for arbitrary topological spaces remains an open question, the topological fundamental group is always a homogeneous space.

It is a well-known result of the covering groups that a subgroup G of the fundamental group at the identity of a semilocally simply connected topological group determines a covering morphism of topological groups with characteristic group G . In this paper we generalize this result to a large class of algebraic objects called topological groups with operations, including topological groups. We ...