Diagonal structures on topological spaces

Abstract

We investigate when a topological space admits a partial product operation satisfying some rather weak continuity restrictions and almost nothing else-the only algebraic requirement is that some element e of X is a left and a right identity with respect to this multiplication. The operation is called partial diagonalization of X at e. Several sufficient conditions for a space to be partially diagonalizable are established. On the other hand, it is shown that certain deep results about the topological structure of compact topological groups can be extended to partially diagonalizable compact spaces. We also discover that partial diagonalizability plays an important role in the theory of cardinal invariants, in the study of homogeneous spaces, and in such classical topics of general topology as the theory of Stone–Čech compactification and the theory of Hewitt–Nachbin compactification. The notions of a Moscow space and of a C-embedding are instrumental in our study.