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.