Editing Support (mathematics) (section)

==Family of supports==

An abstract notion of '''{{em|{{visible anchor|family of supports}}}}''' on a [[topological space]] <math>X,</math> suitable for [[sheaf theory]], was defined by [[Henri Cartan]]. In extending [[Poincaré duality]] to [[manifold]]s 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 <math>\Phi</math> of closed subsets of <math>X</math> is a {{em|family of supports}}, if it is [[down-closed]] and closed under [[finite union]]. Its {{em|extent}} is the union over <math>\Phi.</math> A {{em|paracompactifying}} family of supports that satisfies further that any <math>Y</math> in <math>\Phi</math> is, with the [[subspace topology]], a [[paracompact space]]; and has some <math>Z</math> in <math>\Phi</math> which is a [[Neighbourhood (topology)|neighbourhood]]. If <math>X</math> is a [[locally compact space]], assumed [[Hausdorff space|Hausdorff]], the family of all [[compact subset]]s satisfies the further conditions, making it paracompactifying.