On topological fundamental groups of quotient spaces

Abstract

Let $p:X\to X/A$ be a quotient map, where $A$ is a subspace of $X$. We explore conditions under which $p_{\ast }({\pi }_1^{qtop}(X,x_0))$ is dense in ${\pi }_1^{qtop}(X/A,\ast ))$, where the fundamental groups enjoy the natural quotient topology inherited from the loop space and $p_{\ast }$ is the induced continuous homomorphism by the quotient map $p$. Also, we give some applications to find out some properties for ${\pi }_1^{qtop}(X/A,\ast )$. In particular, we give some conditions in which ${\pi }_1^{qtop}(X/A,\ast )$ is an indiscrete topological group.

13 pages, 1 figure, to appear in Ukrainian Mathematical Journal

References in corpus (3)

• Topological fundamental groups and small generated coverings
• Spanier spaces and covering theory of non-homotopically path Hausdorff spaces
• On H-groups and their applications to topological homotopy groups

Cited by in corpus (1)

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