#### Title

Group Topologies Coarser than the Isbell Topology

#### Document Type

Article

#### Publication Date

2011

#### Publication Title

Topology and its Applications

#### DOI

10.1016/j.topol.2011.06.038

#### Abstract

The Isbell, compact-open and point-open topologies on the set C(X,R) of continuous real-valued maps can be represented as the dual topologies with respect to some collections α(X) of compact families of open subsets of a topological space *X *. Those α(X) for which addition is jointly continuous at the zero function in Cα(X,R) are characterized, and sufficient conditions for translations to be continuous are found. As a result, collections α(X) for which Cα(X,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 Cα(X,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.

#### Recommended Citation

Dolecki, Szymon, Francis Jordan, Frédéric D. Mynard.
2011.
"Group Topologies Coarser than the Isbell Topology."
*Topology and its Applications*, 158 (15): 1962-1968.
doi: 10.1016/j.topol.2011.06.038

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/273