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

Abstract

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 ).