Vol. 12, No. 3, 2019

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A countable space with an uncountable fundamental group

Jeremy Brazas and Luis Matos

Vol. 12 (2019), No. 3, 381–394
Abstract

Traditional examples of spaces that have an uncountable fundamental group (such as the Hawaiian earring space) are path-connected compact metric spaces with uncountably many points. We construct a T0 compact, path-connected, locally path-connected topological space H with countably many points but with an uncountable fundamental group. The construction of H, which we call the “coarse Hawaiian earring” is based on the construction of the usual Hawaiian earring space = n1Cn where each circle Cn is replaced with a copy of the four-point “finite circle”.

Keywords
fundamental group, countable topological space, finite topological space, Hawaiian earring, coarse Hawaiian earring
Mathematical Subject Classification 2010
Primary: 54D10, 55Q52
Secondary: 57M05, 57M10
Milestones
Received: 8 April 2017
Revised: 12 June 2018
Accepted: 9 September 2018
Published: 14 December 2018

Communicated by Józef H. Przytycki
Authors
Jeremy Brazas
Department of Mathematics
West Chester University
West Chester, PA
United States
Luis Matos
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States