The Vietoris mapping theorem for bicompact spaces. II.

Abstract

As a means of proving the isomorphism of the homology groups of two spaces, Vietoris' well-known theorem [5]' has many applications in topology. In this paper we show that this theorem can be extended from the case of compact metric spaces to that of bicompact Hausdorff spaces. Since Vietoris' original proof is somewhat condensed, and also contains one minor slip, we present our proof in some detail. The statement of the theorem, for an arbitrary coefficient group, is found in Section 3 below, as well as a simpler form of the theorem which holds when the coefficient group is suitably restricted. Sections 1 and 2 are devoted to preliminary material, and Section 4 to three lemmas which are used, in the last two sections, in the proofs of the two forms of the theorem.