John Lee,Smooth mfds exercise 2.16

  • Context: Graduate 
  • Thread starter Thread starter seydunas
  • Start date Start date
  • Tags Tags
    Exercise
Click For Summary
SUMMARY

The discussion focuses on Exercise 2.16 from John Lee's book "Smooth Manifolds," which addresses the proof that a topological manifold X is paracompact if every open cover O has a subordinate partition of unity. Participants suggest using the properties of partition of unity to construct a locally refined open cover. Key points include the necessity of understanding the definitions of partition of unity and the locally finite condition, as well as the significance of the support of continuous functions in this context.

PREREQUISITES
  • Understanding of topological manifolds
  • Familiarity with partition of unity in topology
  • Knowledge of paracompactness in topological spaces
  • Basic concepts of continuous functions and their supports
NEXT STEPS
  • Study the definition and properties of partition of unity in topology
  • Research the concept of paracompactness and its implications
  • Learn about locally finite open covers and their significance
  • Explore continuous functions and their support in the context of topology
USEFUL FOR

Mathematicians, particularly those studying topology and differential geometry, as well as students working on exercises related to smooth manifolds and paracompactness.

seydunas
Messages
39
Reaction score
0
Hi,

i have tried to solve exercise2.16 in Lee' s book, smooth manifolds: Let X be a top. mfd with the propert that for every open cover O there exist partition of unity subordinate to O. Show X is paracompact.

I think there is a trick to construct locally refinement open cover by using the conditions of partition of unity. But i can not find it. Can you help me?
 
Physics news on Phys.org
I don't even remember the definitions, but isn't this trivial? re-read the definition of partition of unity and see if it doesn't involve a locally finite condition. remember the set where a continuous function does not vanish is open. i believe this is a tautology, i.e. like proving A+B implies B. see if you agree.
 
Last edited:
If we can take the interior of support coming from partition of unity, we are done.
 

Similar threads

Graduate Regarding fibrations between smooth manifolds
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Manifolds - Charts on Real Projective Spaces
  • · Replies 8 ·
Replies
8
Views
3K
Graduate Local parameterizations and coordinate charts
  • · Replies 14 ·
Replies
14
Views
4K
  • Poll Poll
Geometry Introduction to Smooth Manifolds by Lee
  • · Replies 10 ·
Replies
10
Views
8K
Graduate Wald Lemma 8.1.1-help with proof
  • · Replies 4 ·
Replies
4
Views
3K
  • Poll Poll
Analysis Real and Functional Analysis by Lang
  • · Replies 1 ·
Replies
1
Views
6K
  • Poll Poll
Geometry Algebraic Curves and Riemann Surfaces by Miranda
  • · Replies 1 ·
Replies
1
Views
5K
Graduate Criticism of LQG - What do we miss?
  • · Replies 31 ·
2
Replies
31
Views
7K
Challenge Math Challenge - December 2021
  • · Replies 93 ·
4
Replies
93
Views
15K
Challenge Math Challenge - November 2018
  • · Replies 175 ·
6
Replies
175
Views
27K