Abstract
In 1970, Day and Kelly characterized exponential spaces by a condition (C). Eight years later, Hofmann and Lawson pointed out that this is equivalent to quasi-local compactness, i.e. every neighborhood V of a point contains a smaller one W such that any open cover of V admits a finite subcover of W. These characterizations work with topologies on topologies and may be felt to be not really elementary. This note instead offers an elementary approach which applies to quotient-reflective subcategories as well and includes a natural generalization of the compact-open topology on function spaces.
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Lowen-Colebunders, E., Richter, G. An Elementary Approach to Exponential Spaces. Applied Categorical Structures 9, 303–310 (2001). https://doi.org/10.1023/A:1011268007097
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DOI: https://doi.org/10.1023/A:1011268007097