General Homology Theory in an Arbitrary Space

In the last years several authors1 have developed a homology theory in a compact metric space. This is the way Alexandroff proceeds: By virtue of the Borel theorem one can cover R by a finite system (1) $${U_{\text{1}}},{U_{\text{2}}},...,{U_k}$$ of open sets2 whose norm (= maximum of diameters of the sets (1)) is smaller than a given positive number. Starting from (1), one can now derive a (abstract) complex N whose vertices a 1, a 2,..., a k correspond to the sets (1), and the vertices a v 0, a v 1,..., a v n determine an n-simplex of N if and only if the corresponding sets U v 0 ,U v 1 ,..., U v n have a common point. Then one considers a sequence of coverings of the type (1) whose norms tend to zero, each covering of the sequence being obtained from the preceding one by a subdivision, and one forms the sequence (2) $${N_{\text{1}}},{N_{\text{2}}},{N_3},...$$ of the corresponding complexes (Projektionsfolge). Each cycle C v+1 n situated in N v+1 determines a cycle C v n = πC v+1 n in N v , the projection of C v+1 n . A sequence of cycles contained in the complexes (2) constitutes a cycle in R (Projektionszyklus, Vollzyklus) if there is for each v in N v the homology C v n ~ πC v+1 n .