# On the Topological Fundamental Groups of Quotient Spaces

### Abstract

Let*p*:

*X*→

*X/A*be a quotient map, where

*A*is a subspace of

*X*. We study the conditions under which

*p*

_{∗}(π

_{1}

^{qtop}(

*X*,

*x*

_{0})) is dense in π

_{1}

^{qtop}(

*X*/

*A*,∗)), where the fundamental groups have the natural quotient topology inherited from the loop space and

*p*

_{*}is a continuous homomorphism induced by the quotient map

*p*. In addition, we present some applications in order to determine the properties of π

_{1}

^{qtop}(

*X*/

*A*,∗). In particular, we establish conditions under which π

_{1}

^{qtop}(

*X*/

*A*,∗) is an indiscrete topological group.

Published

25.12.2013

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 65, no. 12, Dec. 2013, pp. 1700–1711, http://umj.imath.kiev.ua/index.php/umj/article/view/2547.

Issue

Section

Research articles