Topological ultraproducts: When is the quotient mapping closed?

Abstract

In the article "Ultraproducts in topology" (General Topology Appl. 7 (1977) 283-308) Paul Bankston investigated ultraproducts of topological spaces (i.e., reduced box products of the form □ u X α , where u ⊂ P(κ) is an ultrafilter) and asked when the quotient map q : □X α → □ u X α is closed (Problem 10.3). We consider more general products-reduced products and prove (in ZFC) that if the X α 's belong to a wide class of spaces, then the mapping q is not closed. Also, we construct some nontrivial examples of ultraproducts such that the map q is closed and give an example of an ultraproduct such that the closeness of q is a statement independent of ZFC. © 1998 Elsevier Science B.V.