The concept of an exponent in category theory generalizes the notion of a set offunctions from one set to another. We first present cartesian closed categories, where exponents always exist. We continue by showing that the category of small categoriesand the category of presheaves are cartesian closed. We then show that the category of topological spaces is not cartesian closed, as exponents do not exist in general.Finally we discuss the category of compactly generated spaces, which represents a natural cartesian closed subcategory of the category of topological spaces.
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