Milnor–Thurston homology groups of the Warsaw Circle

Milnor–Thurston homology theory is a construction of homology theory that is based on measures. It is known to be equivalent to singular homology theory in case of manifolds and complexes. Its behaviour for non-tame spaces is still unknown. This paper provides results in this direction. We prove that Milnor–Thurston homology groups for the Warsaw Circle are trivial except for the zeroth homology group which is uncountable-dimensional. Additionally, we prove that the zeroth homology group is non-Hausdorff for this space with respect a natural topology that was proposed by Berlanga.