Which ordinals have coarser connected Urysohn topologies?
by
William Fleissner
University of Kansas
Coauthors: Jack Porter, University of Kansas; Judith Roitman, University of Kansas

A space is called Urysohn if distinct points have disjoint closed neighborhoods. Every ordinal is homeomorphic to an ordinal of the form w^x·m or of the form w^x·m +w^z, where m < w, z < x, and we use ordinal arithmetic. If z = 0, the ordinal is a compact Hausdorff space, and hence has no coarser connected Hausdorff topology. If a has the form w^x·m or the form w^x·m +w^z, where |w^x·m| ≤ 2^|wz|, then a has a coarser connected Hausdorff topology. Otherwise, not. (In our 2002 Topology and Applications paper.)

This situation for coarser connected Urysohn topologies is not completely known. Partial results show that the conditions are more complicated than in the Hausdorff case. First, it is necessary that a have cofinality w. Second, ordinals of cofinality w of the form w^x·m +w^z with |w^x·m| = |w^z| may or not not have coarser connected Urysohn topologies. To illustrate, let c be the cardinal of the continuum, and let k = (2^c)⁺. The ordinal c·c + c·w has a coarser connected Urysohn topology. The analogous ordinal, k·k+ k·w, does not. However, k^w·k^w + k^w·w does have a coarser connected Urysohn topology.

Date received: February 27, 2006