Embedding partially ordered spaces in topological semilattices

Lloyd D. Tucker
1972 Proceedings of the American Mathematical Society  
A partial order F on a compact space S is called continuous if T is a closed subset of S x S. In this paper, we define and study an embedding of the arbitrary compact continuously partially ordered space (S, T) into a corresponding compact topological semilattice Sr-We show that the structure of entirely determines the structure of (S, T). We prove that the inverse images under O of components in Sr are the order components of (5, T), where elements a and b of 5 are defined to be in the same
more » ... er component of (5, V) if there exists no continuous monotonic map f:(S, r)-»{0, 1} which separates a and b. Finally, we show that Sr is connected if and only if (5, T) has only one order component.
doi:10.1090/s0002-9939-1972-0292724-7 fatcat:gwngztsexbhl5ojjffnd4gepsy