OIF Spaces
by
Joe Mashburn
University of Dayton
Coauthors: Zoltan Balogh, Harold Bennett, Dennis Burke, Gary Gruenhage, David Lutzer

A natural type of base for a topological space is a base B having the following property. For every B Î B the set {A Î B:B Í A} is finite. Such a base is called an OIF (open-in-finite) base, and a space having such a base is called an OIF space. In this talk, basic properties of OIF spaces are explored. We find, for example, that while OIF spaces are productive, not all factors of an OIF product need be OIF, and OIF spaces are not preserved by open perfect mappings. There are T₂ spaces having dense subsets that are not OIF. It is unknown whether there are T₃ spaces with the same properties. Finally, we show that every space X is a closed subset of an OIF space having similar separation properties, and that if X is T₁, it is a G_d subset.

A stronger type of base is a strong OIF base, or SOIF base. A base B for a space X is called an SOIF base if and only if for every A Í X, B restricted to A is an OIF base. For T₂ spaces, SOIF bases and uniform bases are the same. In between OIF spaces and spaces with SOIF bases are OIF spaces in which every subspace is an OIF space. Such spaces are called hereditary OIF spaces, or HOIF spaces. An HOIF space that is compact T₂, or countable compact T₃, or locally compact T₂, will be metrizable.

OIF Spaces

Date received: January 27, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caca-17.