A countable space with an uncountable fundamental group (Q1731020)

The main result of this self contained paper is the construction of a countable space, whose fundamental group is uncountable. \par The construction is based on the following idea. A prototype of a compact space with an uncountable fundamental group is the Hawaiian earring, i.e., the union of a collection of circles of radius $1/n$ sharing a common point. The authors first describe a finite \(T_0\) space on four points, whose fundamental group is $\mathbb{Z}$. They then replace each circle in the Hawaiian earring with this finite model and appropriately adjust the topology on the union, which at this point consists of countably many points. The description of this idea in the paper contains a nice introduction to finite \(T_0\) spaces.