Type of Document Dissertation Author Eklund, Anthony D. URN etd-06092012-141053 Title The fine topology and other topologies on C(X,Y). Degree PhD Department Mathematics Advisory Committee

Advisor Name Title McCoy, Robert A. Committee Chair Aull, C. E. Committee Member Greenberg, William Committee Member Johnson, Lee W. Committee Member Parry, Charles J. Committee Member Keywords

- Topology
Date of Defense 1978-05-05 Availability restricted Abstract"The Fine Topology" C(X,Y) where (Y,d) is a metric space is referred to, in an exercise in [14], as the topology generated by basic open neighborhoods of the form B(f,E) = {g: d(f(x),g(x)) < E(x)} where E is a positive continuous real valued function. So in the fine topology, a function g is close to f if g(x) is continuously close to f(x); whereas in the uniform topology, g(x) must be uniformly close to f(x), that is, within a constant distance of f(x). So the fine topology is an obvious refinement of the uniform topology. This topology has not been extensively studied before, and it is the purpose of this paper to see how the fine topology fits in with the lattice of other well studied topologies on C(X,Y), and to study some properties of this topology in itself. Furthermore, other results on these well studied topologies will-be examined and compared with the fine topology.

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