On the compact open and finest splitting topologies

Abstract

In [M.H. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105–145] it is shown that in the set C ( N ω , N ) of all continuous maps of N ω into N, where N is an infinitely countable discrete topological space, the compact-open topology is not the finest splitting topology. Since N ω is consonant (see [S. Dolecki, G.H. Greco, A. Lechicki, When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide? Trans. Amer. Math. Soc. 347 (1995) 2869–2884]) the Isbell topology on C ( N ω , N ) also is not the finest splitting topology. This result is generalized in the present paper proving that it is true also for spaces having the so-called Specific Extension Property. The following spaces have the Specific Extension Property: (a) infinitely countable free unions of non-empty spaces, (b) non-compact Lindelöf zero-dimensional spaces, and (c) metric locally convex linear spaces. In particular, we prove that on the set of all real-valued functions on the (separable infinite dimensional) Hilbert space the compact-open topology does not coincide with the finest splitting topology.