On locally 1-connectedness of quotient spaces and its applications to fundamental groups

Abstract

Let $X$ be a locally 1-connected metric space and $A_1,A_2,...,A_n$ be connected, locally path connected and compact pairwise disjoint subspaces of $X$. In this paper, we show that the quotient space $X/(A_1,A_2,...,A_n)$ obtained from $X$ by collapsing each of the sets $A_i$'s to a point, is also locally 1-connected. Moreover, we prove that the induced continuous homomorphism of quasitopological fundamental groups is surjective. Finally, we give some applications to find out some properties of the fundamental group of the quotient space $X/(A_1,A_2,...,A_n)$.

11 pages

References in corpus (4)

• The fundamental group as a topological group
• The topological fundamental group and free topological groups
• Discreteness and Homogeneity of the Topological Fundamental Group
• On topological fundamental groups of quotient spaces