Atlas: Volterra Spaces by David Gauld

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The Eleventh Summer Conference on General Topology and Applications
August 10-13, 1995
University of Southern Maine
Gorham, ME, USA

Organizers
J. Baumgartner, D. Briggs, J. deBakker, B. Flagg, G. Gruenhage, M. Guay, Y. Kong, R. Kopperman, S. Shore, J. Rutten, J. Vaughan

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Volterra Spaces
by
David Gauld
University of Auckland

In 1881 at the age of 19, Vito Volterra proved the following:

Let f : R --> R be a function for which C(f) and D(f) are dense. Then there is no function g : R --> R such that C(f) = D(g) and D(f) = C(g).

By C(f), respectively D(f), we mean the set of points at which f is continuous, respectively discontinuous.

In this talk joint results with Sina Greenwood and Zbiggie Piotrowski will be presented which show that for any topological space X the following two conditions are equivalent:

for each pair f, g : X --> R for which C(f) and C(g) are dense in X the set C(f) \cap C(g) is dense (respectively is non-empty);

for each developable space Y and each pair f, g : X --> Y for which C(f) and C(g) are dense in X the set C(f) \cap C(g) is dense (respectively is non-empty).

Note that Volterra's result is equivalent to the ``non-empty" version of the first statement for R.

The question where we have a family of functions will also be explored.

Date received: April 12, 1996