1137: Partition of Unity Subordinate to Open Cover of Topological Space

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definition of partition of unity subordinate to open cover of topological space

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Topics

About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of Euclidean topological space.
• The reader knows a definition of support of map from topological space into ring or module.
• The reader knows a definition of locally finite set of subsets of topological space.

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Target Context

• The reader will have a definition of partition of unity subordinate to open cover of topological space.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$T$: $\in \{\text{ the topological spaces }\}$
$\mathbb{R}$: $=\text{ the Euclidean topological space }$
$\{U_j|j\in J\}$: $\in \{\text{ the open covers of }T\}$
$\ast \{{\rho }_j|j\in J\}$: ${\rho }_j:T\to \mathbb{R}$, $\in \{\text{ the continuous maps }\}$
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Conditions:
1) $\mathrm{\forall }j\in J,\mathrm{\forall }t\in T(0\le {\rho }_j(t)\le 1)$
$\wedge$
2) $\mathrm{\forall }j\in J(supp{\rho }_j\subseteq U_j)$
$\wedge$
3) $\{supp{\rho }_j|j\in J\}\in \{\text{ the locally finite sets of subsets of }T\}$
$\wedge$
4) $\mathrm{\forall }t\in T(\sum _{j\in J}{\rho }_j(t)=1)$
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2: Note

$\sum _{j\in J}{\rho }_j(t)$ makes sense because $\{supp{\rho }_j|j\in J\}$ is a locally finite set of subsets: for each $t$, there are only some finite ${\rho }_j(t)$ s that are nonzero, so, $\sum _{j\in J}{\rho }_j(t)$ is practically a finite sum.

When $T$ is a $C^{\mathrm{\infty }}$ manifold with boundary and ${\rho }_j$ s are $C^{\mathrm{\infty }}$, $\{{\rho }_j|j\in J\}$ is called "$C^{\mathrm{\infty }}$ partition of unity".

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References

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