Group topologies coarser than the Isbell topology

Abstract

The Isbell, compact-open and point-open topologies on the set $C(X,\mathbb{R})$ of continuous real-valued maps can be represented as the dual topologies with respect to some collections $\alpha(X)$ of compact families of open subsets of a topological space $X$. Those $\alpha(X)$ for which addition is jointly continuous at the zero function in $C_\alpha(X,\mathbb{R})$ are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections $\alpha(X)$ for which $C_{\alpha}(X,\mathbb{R})$ is a topological vector space are defined canonically. The Isbell topology coincides with this vector space topology if and only if $X$ is infraconsonant. Examples based on measure theoretic methods, that $C_\alpha (X,\mathbb{R})$ can be strictly finer than the compact-open topology, are given. To our knowledge, this is the first example of a splitting group topology strictly finer than the compact-open topology.