An abstract notion of family of supports on a topological space X, suitable for sheaf theory, was defined by Henri Cartan. In extending Poincaré duality to manifolds that are not compact, the 'compact support' idea enters naturally on one side of the duality; see for example Alexander-Spanier cohomology.
Bredon, Sheaf Theory (2nd edition, 1997) gives these definitions. A family Φ of closed subsets of X is a family of supports, if it is down-closed and closed under finite union. Its extent is the union over Φ. A paracompactifying family of supports satisfies further than any Y in Φ is, with the subspace topology, a paracompact space; and has some Z in Φ which is a neighbourhood. If X is a locally compact space, assumed Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying.
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