Paracompactness and metrizable manifolds

  • Thread starter Thread starter aleazk
  • Start date Start date
  • Tags Tags
    Manifolds
aleazk
Science Advisor
Insights Author
Messages
83
Reaction score
62
Hi, where I can find a proof of the theorem that establishes that a manifold is metrizable (with a riemannian metric) if and only if is paracompact?.
 
Last edited:
Physics news on Phys.org
Wow, I found a proof in Kobayashi and Nomizu, Vol I, and it seems pretty hard :frown:. I think that simply I will have to accept the result :cry:
 
Hi aleazk! :smile:

Try "topology" of Munkres, it gives an easy proof of the result (but the proof is still several pages long :smile: )
 

Thread '2-sphere intrinsic definition by gluing disks' boundaries'

A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
View full post »

Similar threads

Munkres Topology - Chapter 7 - Complete Metric Spaces and Function Spaces
Replies
2
Views
7K
I ##SL(2,\mathbb R)## Lie group as manifold
Replies
12
Views
2K
Foliation of 4-dimensional connected Hausdorff orientable paracompact manifold
Replies
13
Views
5K
Can't understand a detail in paracompactness->normality
Replies
3
Views
1K
I How do charts on differentiable manifolds have derivatives without a metric?
Replies
20
Views
4K
I Differentiable manifolds over fields other than R, C
Replies
13
Views
2K
Idea behind topological manifold definition.
Replies
3
Views
2K
I Show that a "cross" is not a topological manifold
Replies
20
Views
3K
I Pseudo-Riemaniann isometries
Replies
4
Views
2K
I Why Isn't the Intersection of Two Lines a 1D Manifold?
Replies
21
Views
3K

Hot Threads

Recent Insights

Back
Top