The General Topological Theorem of Duality for Closed Sets *

In recent years there has appeared a series of papers devoted to topological theorems of duality. Leaving out the theorems of duality of Poincaré, which, by the way, are closely related to these questions, the general statement of the question is the following: Let M be a manifold and F one of its compact subsets. It is required to study the topological properties of the space M — F, starting from those of the set F. In particular, we may suppose that M is a Euclidean space or, what is essentially the same, a sphereical manifold, F is homeomorphic to some complex, and the topological properties studied are the Betti numbers modulo two. In this form the problem was solved by Alexander. ¹ Namely, he showed that the r-dimensional Betti number modulo two of the space M — F equals the (n − r − 1)-dimensional Betti number of the complex F, where n is the dimensionality of the spherical manifold M.