On Locally 1-Connectedness of Quotient Spaces and Its Applications to Fundamental Groups

Ali Pakdaman, Hamid Torabi, Behrooz Mashayekhy

Let $X$be a locally 1-connected metric space and $A_1,A_2,\ldots,A_n$ be connected, locally path connected and compact pairwise disjoint subspaces of $X$. In this paper, we show that the quotient space $X/(A_1,A_2,\ldots,A_n)$ 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 $X/(A_1,A_2,\ldots,A_n)$. the quotient space