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Quasitopological space
A set X equipped with a function that associates to every compact Hausdorff space K a collection of maps K→C satisfying certain natural conditions
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In mathematics, a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions. A set with a quasi-topology is called a quasitopological space.

They were introduced by Spanier, who showed that there is a natural quasi-topology on the space of continuous maps from one space to another.