Topological ultraproducts: when is the quotient mapping closed?

✍ Scribed by MilošS. Kurilić

• Book ID: 104295286
• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 440 KB
• Volume: 87
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(97)00132-6

✦ Synopsis

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 0,X,, where L4 C P(n) is an ultrafilter) and asked when the quotient map q: OX, + 0,X, is closed (Problem 10.3). We consider more general products-reduced products and prove (in ZFC) that if the X,'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.