Partition of Unity
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High Quality Content by WIKIPEDIA articles! Partitions of unity are useful because they often allow one to extend local constructions to the whole space. The existence of partitions of unity assumes two distinct forms: 1. Given any open cover {Ui}i I of a space, there exists a partition { i}i I indexed over the same set I such that supp i Ui. Such a partition is said to be subordinate to the open cover {Ui}i. 2. Given any open cover {Ui}i I of a space, there exists a partition { j}j J indexed over a possibly distinct index set J such that each j has compact support and for each j J, supp j Ui ...
High Quality Content by WIKIPEDIA articles! Partitions of unity are useful because they often allow one to extend local constructions to the whole space. The existence of partitions of unity assumes two distinct forms: 1. Given any open cover {Ui}i I of a space, there exists a partition { i}i I indexed over the same set I such that supp i Ui. Such a partition is said to be subordinate to the open cover {Ui}i. 2. Given any open cover {Ui}i I of a space, there exists a partition { j}j J indexed over a possibly distinct index set J such that each j has compact support and for each j J, supp j Ui for some i I. Thus one chooses either to have the supports indexed by the open cover, or the supports compact. If the space is compact, then there exist partitions satisfying both requirements.
