Cohomology and Dimension of Nearness Spaces

Abstract

It is a well-known fact that cohomology theory leads to better results in dimension theory than homology theory. The beautiful results characterizing finite-dimensional compact metric spaces by means of homology resp. cohomology (cf. Hurewicz and Wallman [48]) may be generalized in a slightly modified form to compact Hausdorff spaces provided Lebesgue’s covering dimension is considered (cf. Nagata [64]). But already for the wider class of paracompact Hausdorff spaces a corresponding homological characterization of covering dimension is not valid. Nevertheless a cohomological characterization of finite-dimensional paracompact Hausdorff spaces is known. In 1952 C.H. Dowker [24] has shown that Čech’s cohomology theory (and homology theory) may be defined for structures which — as we know today — include nearness structures. H.L. Bentley [11] and D. Czarcinski [21] have proved that these theories satisfy a variant of the Eilenberg-Steenrod axioms. During this chapter it is expected that the reader is acquainted with simplicial cohomology and classical Čech cohomology.