Eilenberg in 1944 ([E])2 removed some of the difficulties associated with the classical concept of cell and obtained a new singular homology theory. In 1948, Hurewicz, Dugundji, and Dowker ([H]) applied the methods of inverse and direct limit groups in conjunction with mappings of polytopes into a space to obtain another singular homology theory. The primary purpose of this paper is to show that these two methods yield isomorphic cohomology (homology) groups. We prove this result by constructing a simplicial polytope P(X) which is a geometric realization of the Eilenberg singular complex of the space X, and then showing that the n-cohomology group of P(X) is isomorphic to the n-cohomology groups of X as defined by Eilenberg and by Hurewicz, Dugundji, and Dowker. We also prove that the Hurewicz homotopy groups of P(X) and the space X are isomorphic. The above results are stated in full in Section 6. Section 6 contains the statements of the principal results of this paper, although Theorem I is given in Section 5 and Theorem VII is given in Section 10. In Sections 7 to 9 we prove the results of Section 6. In Sections 2 to 4 we briefly summarize the definitions and theorems from the papers of J. H. C. Whitehead ([W]); Hurewicz, Dugundji, and Dowker ([H]); and Eilenberg ([E]) which are needed in the remainder of this paper. The basic notation used will be that of [HW]. Beyond that, our notation will be consistent with [H], and, in regard to the Eilenberg complex, [E]. We will use to denote group isomorphisms, and to denote homotopy relations. The topological spaces and groups considered here are of the most general type with no separation axioms postulated. We will use infinite cochains over a topological, abelian coefficient group. Chains will be finite and with a discrete, abelian coefficient group. The word simplex will always mean a closed geometric (Euclidean) simplex unless we explicitly state otherwise. We will let iz-n(Y, y) denote the Hurewicz n-homotopy group of the space Y with the base point