In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — the extension of a concept, idea, or sign consists of the things to which it applies, in contrast with its comprehension or intension, which consists very roughly of the ideas, properties, or corresponding signs that are implied or suggested by the concept in question.
In philosophical semantics or the philosophy of language, the 'extension' of a concept or expression is the set of things it extends to, or applies to, if it is the sort of concept or expression that a single object by itself can satisfy.
The extension of a whole statement, as opposed to a word or phrase, is defined (since Gottlob Frege's "On Sense and Reference") as its truth value.
This kind of extension is used so constantly in contemporary mathematics based on set theory that it can be called an implicit assumption.
If not, only existing, actual things can be in the extension of a concept or expression.
(Sherlock Holmes seems to be an actual example of a fictional character; one might think there are many other characters Arthur Conan Doyle might have invented, though he actually invented Holmes.)
Some fundamental formulations in the field of general semantics rely heavily on a valuation of extension over intension.