definition of partition of unity subordinate to open cover of topological space
Topics
About: topological space
The table of contents of this article
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.
Target Context
- The reader will have a definition of partition of unity subordinate to open cover of topological space.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( T\): \(\in \{\text{ the topological spaces }\}\)
\( \mathbb{R}\): \(= \text{ the Euclidean topological space }\)
\( \{U_j \vert j \in J\}\): \(\in \{\text{ the open covers of } T\}\)
\(*\{\rho_j \vert j \in J\}\): \(\rho_j: T \to \mathbb{R}\), \(\in \{\text{ the continuous maps }\}\)
//
Conditions:
1) \(\forall j \in J, \forall t \in T (0 \le \rho_j (t) \le 1)\)
\(\land\)
2) \(\forall j \in J (supp \rho_j \subseteq U_j)\)
\(\land\)
3) \(\{supp \rho_j \vert j \in J\} \in \{\text{ the locally finite sets of subsets of } T\}\)
\(\land\)
4) \(\forall t \in T (\sum_{j \in J} \rho_j (t) = 1)\)
//
2: Note
\(\sum_{j \in J} \rho_j (t)\) makes sense because \(\{supp \rho_j \vert j \in J\}\) 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 \(C^\infty\) manifold with boundary and \(\rho_j\) s are \(C^\infty\), \(\{\rho_j \vert j \in J\}\) is called "\(C^\infty\) partition of unity".
