On the coincidence of zeroth Milnor-Thurston homology with singular   homology
  In this paper we prove that the zeroth Milnor-Thurston homology group coincides with singular homology for Peano Continua. More- over, we show that the canonical homomorphism between these ho- mology theories may not be injective. However, it is proved that it is injective when a space has Borel path-components. 
Janusz Przewocki, Andreas Zastrow
math.GT math.KT