Aspects of the embeddability ordering in topology
by
Michael Gormley
The Queen's University of Belfast
Coauthors: T. B. M. McMaster

Let T(\alpha) = {T_i: i in I} be the set of homeomorphism classes of topologies definable on a cardinal number \alpha. The quasi-order on T(\alpha) is defined as follows:
T_i < = T_j iff (\alpha, T_i) \hookrightarrow (\alpha, T_j), that is (\alpha, T_i) embeds into (\alpha, T_j).

We investigate some of the properties that T(\alpha) (thus ordered) possesses. For example, we know that every pair of incomparable members fails to have a least upper bound. We will exhibit two points T₁ and T₂, which have an upper bound T₃, such that there does not exist a minimal upper bound T₄ (to T₁ and T₂) with T₄ < = T₃. That is to say, T(\alpha) does not form an upper semi-multilattice. This involves defining a topology on the union of a collection of pairwise disjoint topological spaces indexed by a poset. (25 minute talk.)

Date received: August 23, 2000