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International Journal of Mathematics and Mathematical Sciences
Volume 12 (1989), Issue 1, Pages 1-8
http://dx.doi.org/10.1155/S0161171289000013

Nowhere dense sets and real-valued functions with closed graphs

Department of Mathematics, University of Alberta, Edmonton T6G 2G1, Alberta, Canada

Received 19 April 1988; Revised 10 May 1988

Copyright © 1989 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

Closed and nowhere dense subsets which coincide with the points of discontinuity of real-valued functions with a closed graph on spaces which are not necessarily perfectly normal are investigated. Certain Gδ subsets of completely regular and normal spaces are characterized. It is also shown that there exists a countable connected Urysohn space X with the property that no closed and nowhere dense subset of X coincides with the points of discontinuity of a real-valued function on X with a closed graph.