An infraconsonant nonconsonant completely regular space

Let R denote the real numbers. We construct in ZFC a countable space X such that X has exactly one non-isolated point, X is infraconsonant, and X is not consonant. We conclude that X is a completely regular space such that Isbell topology on C ( X , R ) is a group topology that coincides with the natural (finest splitting) topology on C ( X , R ) , but the Isbell and compact-open topologies on C ( X , R ) do not coincide. The example answers two open problems in the literature.