Diagonalizable spaces
by
Peter Nyikos
University of South Carolina

In the first issue of Applied General Topology, Arhangel'skii called a space with a binary operation a semitopoid if the operation is separately continuous and a it topoid if it is jointly continuous. He also called a space X diagonalizable at e [resp. continuously diagonalizable at e] if e can be an identity element in a binary operation on X which at which the operation is separately [resp. jointly] continuous. We discuss some theorems and examples involving these concepts. including some of Arhangel'skii's many results, as well as:

Theorem. If X is a space with a singleton {e} which is the intersection of a chain of closed neighborhoods of e, then X is continously diagonalizable at e. If {e} is the intersection of a countable collection of clopen sets, then X can be made into a topological semigroup with identity e.

Examples. The long ray can be made into a topological semigroup with identity, but the long line cannot even be made into a semitopoid with identity.

This contrast is reminiscent of the well-known contrast between S¹ and S² - the former is a topological group, while the latter cannot even be made into a topological loop.

Problem. Can S² be made into a [semi-]topoid with identity?

Date received: May 31, 2003