1. A theorem which asserts that for a given space, the (n - p ) dimensional homology group is isomorphic to a p-dimensional cohomology group for each p = 0, ..., n, provided certain conditions are met.

2. Let G be either a compact group or a discrete group, let X be its character group, and let G' be the character group of X; then there is an isomorphism of G onto G' so that the groups G and G' may be identified.

3. If either of two dual linear-programming problems has a solution, then so does the other.