Topologies on the Space of Continuous Functions
by
L'. Holá
Coauthors: G. Di Maio, D. Holý, R. A. McCoy

Let X be a Tychonoff space and (X, d) be a metric space. Denote by C(X, Y) the space of continuous functions from X to Y. We study the fine topology on C(X, Y). The fine topology has many applications in approximation theory and differential geometry (M. Hirsch, Differential Topology, Springer Verlag, 1976 and E. Hewitt, Trans. Amer. Math. Soc. (64), 1943, 54 - 99).

If X is a countably paracompact normal space, then the fine topology coincides with the graph topology and if moreover Y is separable, then the fine topology coincides with the Krikorian topology. So under these conditions on X and Y the fine topology is independent from the compatible metric d on Y.

We study also cardinal invariants of the fine topology on C(X, R), where R is the space of reals. The fine topology on C(X, R) is seen to behave like a metric topology in the sense that weight, density, Lindelof number and cellularity are all equal for this topology.

To answer some questions from the paper ``G. DiMaio, S. Naimpally, Proximal Topologies 10, Q & A in General Topology (1992), 97 - 125'' other function spaces are investigated, namely, Krikorian topology, open-cover topology, graph topology, topology of uniform convergence and proximal graph topology.

Date received: June 24, 1996

Copyright © 1996 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 # caah-38.