A countable space with an uncountable fundamental group

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”.