# When is the Isbell Topology a Group Topology?

## Document Type

Article

## Publication Date

2010

## Publication Title

Topology and Its Applications

## DOI

10.1016/j.topol.2009.06.019

## Abstract

Conditions on a topological space *X * under which the space C(X,R) of continuous real-valued maps with the Isbell topology *κ * is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in Cκ(X,R) if and only if *X * is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in Cκ(X,R) if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if *X* is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces.

## Recommended Citation

Dolecki, Szymon, Frédéric D. Mynard.
2010.
"When is the Isbell Topology a Group Topology?."
*Topology and Its Applications*, 157 (8): 1370-1378.
doi: 10.1016/j.topol.2009.06.019

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/275