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partition of unity (Definition)

Let $ X$ be a topological space. A partition of unity is a collection of continuous functions $ \{\varepsilon_i \colon X \to [0,1]\}$ such that

$\displaystyle \sum_i \varepsilon_i(x) = 1$   $\displaystyle \mbox{for all $x \in X$}$$\displaystyle .$ (1)

A partition of unity is locally finite if each $ x$ in $ X$ is contained in an open set on which only a finite number of $ \varepsilon_i$ are non-zero. That is, if the cover $ \{\varepsilon_i^{-1}((0,1])\}$ is locally finite.

A partition of unity is subordinate to an open cover $ \{U_i\}$ of $ X$ if each $ \varepsilon_i$ is zero on the complement of $ U_i$.

Example 1 (Circle)
A partition of unity for $ \mathbb{S}^1$ is given by $ \{ \sin^2(\theta/2), \cos^2(\theta/2) \}$ subordinate to the covering $ \{ (0, 2\pi), (-\pi, \pi) \}$.

Application to integration

Let $ M$ be an orientable manifold with volume form $ \omega$ and a partition of unity $ \{\varepsilon_i(x)\}$. Then, the integral of a function $ f(x)$ over $ M$ is given by

$\displaystyle \int_M f(x) \omega = \sum_i \int_{U_i} \varepsilon_i(x) f(x) \omega. $
It is independent of the choice of partition of unity.



"partition of unity" is owned by mhale.
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Also defines:  locally finite partition of unity, subordinate to an open cover

Attachments:
existence of partitions of unity (Theorem) by asteroid
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Cross-references: function, integral, volume form, orientable manifold, application, covering, complement, open cover, locally finite, cover, number, open set, contained, finite, continuous functions, collection, unity, topological space
There are 7 references to this entry.

This is version 7 of partition of unity, born on 2003-02-26, modified 2006-09-08.
Object id is 4063, canonical name is PartitionOfUnity.
Accessed 7786 times total.

Classification:
AMS MSC58A05 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Differentiable manifolds, foundations)
 54D20 (General topology :: Fairly general properties :: Noncompact covering properties )

Pending Errata and Addenda
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Discussion
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Definition insufficient by joethelion on 2006-12-10 00:42:31
Surely we can do better than a circular, one-word semantic definition here....try Kelley, (even Wikepedia!)....
To quote someone: 'incomplete truth is better than no information'.

Best wishes.
[ reply | up ]
partition of unity by Hui9 on 2005-01-16 15:06:24

 Could you tell me why there always exists such functions to form partition of unity ?
[ reply | up ]

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