An Ultracoproduct Mapping Theorem, with Applications to Continua
by
Paul Bankston
Marquette University, Milwaukee, WI

The ultracoproduct construction for compact Hausdorff spaces occupies a niche miraculously similar to that occupied by the ultraproduct construction in first-order model theory. One of its uses is to define a hierarchy of continuous maps, in parallel with the one in model theory defined according to the preservation of the truth of formulas of specified quantifier complexity. Maps of level zero correspond to embeddings, maps of level one to existential embeddings, and so forth.

The mapping theorem in the title specifies the existence of a map of level k+1 from an ultracoproduct of members of an inverse system to the limit of that system, when all the connecting maps are of level k. Applications include: (i) the preservation of covering dimension and continuum (in)decomposability under inverse limits; and (ii) the construction of an ultracopower of the unit interval that is neither locally connected nor a hereditarily decomposable continuum.

Date received: January 9, 2002