#### 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 ${T}_{0}$ 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 $ℍ={\bigcup }_{n\ge 1}{C}_{n}$ where each circle ${C}_{n}$ 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