2024-12-01

881: For C Vectors Bundle, C Section Along Closed Subset of Base Space Can Be Extended to Over Whole Base Space with Support Contained in Any Open Neighborhood of Subset

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description/proof of that for C vectors bundle, C section along closed subset of base space can be extended to over whole base space with support contained in any open neighborhood of subset

Topics


About: C manifold

The table of contents of this article


Starting Context



Target Context


  • The reader will have a description and a proof of the proposition that for any C vectors bundle, any C section along any closed subset of the base space can be extended to over the whole base space with the support contained in any open neighborhood of the subset.

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:
(E,M,π): { the d -dimensional C vectors bundles of rank k}
C: { the closed subsets of M}
UC: { the open neighborhoods of C on M}
s: :CME, { the C sections along C}
//

Statements:
s:ME{ the C sections of π}(s|C=ssupp sUC)
//


2: Natural Language Description


For any d-dimensional C vectors bundle of rank k, (E,M,π), any closed subset, CM, and any open neighborhood of C, UCM, any C section along C, s:CE, can be extended to a C section of π, s:ME, such that s|C=s and supp sUC.


3: Proof


Whole Strategy: Step 1: for each point, pC, take an open neighborhood of p, UpM, and a C section, sp:UpE, such that UpUC and sp|UpC=s|CUp; Step 2: take the open cover of M, {Up:pC}{MC}, and a partition of unity subordinate to the cover, {ρp|pC}{ρ0}; Step 3: define sp:ME as ρpsp on Up and 0 on MUp; Step 4: define s:=pCsp; Step 5: see that s satisfies the conditions.

Step 1:

For each point, pC, there are an open neighborhood of p, UpM, and a C section, sp:UpE, such that sp|UpC=s|CUp, by the proposition that for any C vectors bundle and any section from any subset of the base space Ck at any point where 0<k, there is a Ck extension on an open-neighborhood-of-point domain.

Up can be taken to be such that UpUC: if not so, we can take UpUC instead of Up, and C-ness is not harmed by it, by the proposition that for any map between any arbitrary subsets of any C manifolds with boundary Ck at any point, where k includes , the restriction on any domain that contains the point is Ck at the point. So, let us suppose that UpUC.

Step 2:

{Up:pC}{MC} is an open cover of M, and there is a partition of unity subordinate to the cover, {ρp|pC}{ρ0}, by the proposition that for any open cover of any C manifold, there is a C partition of unity subordinate to the cover.

Step 3:

Let us define sp:ME as ρpsp on Up and 0 on MUp.

sp is C, because it is C on Up and is 0 on open Msupp ρp, so, C there, and M=Up(Msupp ρp) as supp ρpUp.

Step 4:

Let us define s:=pCsp, which is well-defined, because at any point, pM, only some finite terms are nonzero, by the definition of partition of unity.

Step 5:

s is C, because for any point, pM, there is an open neighborhood, UpM, on which the sum is a finite sum, which is a sum of some C sections.

For each point, pC, s(p)=pCρp(p)sp(p) for a finite CC, but sp(p)=s(p), so, =pCρp(p)s(p)=(pCρp(p)+ρ0(p))s(p)=s(p), because ρp(p) for pCC and ρ0(p) are 0 anyway. So, s is indeed a C extension of s.

supp spCsupp ρp=pCsupp ρp=pCsupp ρppCUpUC, by the proposition that for any locally finite set of subsets of any topological space, the closure of the union of the subsets is the union of the closures of the subsets.


References


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