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International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 1, Pages 111-116
http://dx.doi.org/10.1155/S0161171293000134

The open-open topology for function spaces

Department of Mathematical Sciences, Saint Mary's College of California, Moraga, CA. 94575, USA

Received 20 February 1991; Revised 22 January 1992

Copyright © 1993 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let (X,T) and (Y,T*) be topological spaces and let FYX. For each UT, VT*, let (U,V)={fF:f(U)V}. Define the set S={(U,V):UT and VT*}. Then S is a subbasis for a topology, T on F, which is called the open-open topology. We compare T with other topologies and discuss its properties. We also show that T, on H(X), the collection of all self-homeomorphisms on X, is equivalent to the topology induced on H(X) by the Pervin quasi-uniformity on X.