Editing Partition of unity (section)

== Existence ==
The existence of partitions of unity assumes two distinct forms:

# Given any [[open cover]] <math>\{ U_i \}_{i \in I}</math> of a space, there exists a partition <math>\{ \rho_i \}_{i \in I}</math> indexed ''over the same set'' {{tmath|I}} such that [[Support (mathematics)|supp]] <math>\rho_i \subseteq U_i.</math> Such a partition is said to be '''subordinate to the open cover''' <math>\{ U_i \}_i.</math>
# If the space is locally compact, given any open cover <math>\{ U_i \}_{i \in I}</math> of a space, there exists a partition <math>\{ \rho_j \}_{j \in J}</math> indexed over a possibly distinct index set {{tmath|J}} such that each {{tmath|\rho_j}} has [[compact support]] and for each {{tmath|j \in J}}, supp <math>\rho_j \subseteq U_i</math> for some {{tmath|i \in I}}.

Thus one chooses either to have the [[support (mathematics)|supports]] indexed by the open cover, or compact supports.  If the space is [[compact space|compact]], then there exist partitions satisfying both requirements.

A finite open cover always has a continuous partition of unity subordinate to it, provided the space is locally compact and Hausdorff.<ref>{{cite book|last=Rudin|first=Walter|title=Real and complex analysis|year=1987|publisher=McGraw-Hill|location=New York|isbn=978-0-07-054234-1|pages=40|edition=3rd}}</ref> 
[[Paracompact space|Paracompactness]] of the space is a necessary condition to guarantee the existence of a partition of unity [[paracompact space|subordinate to any open cover]].  Depending on the [[category (mathematics)|category]] to which the space belongs, this may also be a sufficient condition.<ref>{{cite book|first1=Charalambos D.|last1=Aliprantis|first2=Kim C.|last2=Border
|title=Infinite dimensional analysis: a hitchhiker's guide|year=2007|publisher=Springer|location=Berlin|isbn=978-3-540-32696-0| pages=716|edition=3rd}}</ref> In particular, a compact set in the [[Euclidean space]] admits a smooth partition of unity subordinate to any finite open cover. The construction uses [[mollifier]]s (bump functions), which exist in continuous and [[smooth manifolds]], but not necessarily in [[analytic manifold]]s. Thus for an open cover of an analytic manifold, an analytic partition of unity subordinate to that open cover generally does not exist. ''See'' [[analytic continuation]].

If {{tmath|R}} and {{tmath|T}} are partitions of unity for spaces {{tmath|X}} and {{tmath|Y}} respectively, then the set of all pairs <math>\{ \rho\otimes\tau :\ \rho \in R,\ \tau \in T \}</math> is a partition of unity for the [[cartesian product]] space {{tmath|X \times Y}}. The tensor product of functions act as <math>(\rho \otimes \tau )(x,y) = \rho(x)\tau(y).</math>