Discreteness and Homogeneity of the Topological Fundamental Group

Abstract

For a locally path connected topological space, the topological fundamental group is discrete if and only if the space is semilocally simply-connected. While functoriality of the topological fundamental group for arbitrary topological spaces remains an open question, the topological fundamental group is always a homogeneous space.

arXiv comment8 pages, 2 figures

References in corpus (3)

• The fundamental group of the harmonic archipelago
• The topological Hawaiian earring group does not embed in the inverse limit of free groups
• Topological fundamental groups can distinguish spaces with isomorphic homotopy groups

Cited by in corpus (10)

• The fundamental group as a topological group
• The topological fundamental group and free topological groups
• Topological fundamental groups and small generated coverings
• On H-groups and their applications to topological homotopy groups
• On Topological Shape Homotopy Groups
• Multiplication is discontinuous in the Hawaiian earring group (with the quotient topology)
• On the discontinuity of the ${\pi }_1$-action
• On quasitopological homotopy groups of Inverse Limit Spaces
• On Subgroup Topologies on Fundamental Groups
• On locally 1-connectedness of quotient spaces and its applications to fundamental groups