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aleazk
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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?.
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Paracompactness and metrizable manifolds
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In summary, paracompactness is a property of a topological space that guarantees the existence of a locally finite covering. It is closely related to metrizable manifolds and has important consequences, such as normality and the Whitney embedding theorem. Paracompactness is useful in the study of manifolds because it allows for the construction of partitions of unity, which are essential in many geometric and analytic arguments.
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aleazk
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Wow, I found a proof in Kobayashi and Nomizu, Vol I, and it seems pretty hard 
. I think that simply I will have to accept the result 
. I think that simply I will have to accept the result - #3
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Hi aleazk!
Try "topology" of Munkres, it gives an easy proof of the result (but the proof is still several pages long )
Try "topology" of Munkres, it gives an easy proof of the result (but the proof is still several pages long
)Related to Paracompactness and metrizable manifolds
1. What is paracompactness?
Paracompactness is a property of a topological space that ensures the existence of a certain type of covering, called a locally finite covering, which has important applications in analysis and geometry. A space is said to be paracompact if every open cover has a locally finite open refinement, meaning that each point in the space has a neighborhood that intersects only finitely many sets in the refinement.
2. How does paracompactness relate to metrizable manifolds?
A metrizable manifold is a topological space that is both metrizable (i.e. can be equipped with a metric) and locally Euclidean. Paracompactness is a necessary condition for a manifold to be metrizable. In fact, every metrizable manifold is paracompact, but not every paracompact space is a metrizable manifold.
3. What are some examples of paracompact spaces?
Examples of paracompact spaces include any metric space, any smooth manifold, and any Hausdorff space that is second-countable (i.e. has a countable base). In particular, all Euclidean spaces are paracompact.
4. Are there any consequences of a space being paracompact?
Yes, there are several important consequences of paracompactness. For instance, every paracompact space is normal (i.e. for any two disjoint closed sets, there exists open sets containing each set that are also disjoint), which is a desirable property for spaces in analysis. Additionally, paracompactness is often used in the proof of the Whitney embedding theorem, which states that every smooth manifold can be smoothly embedded in some Euclidean space.
5. How is paracompactness useful in the study of manifolds?
Paracompactness is a crucial tool in the study of manifolds because it allows for the construction of partitions of unity, which are essential in many geometric and analytic arguments. These partitions of unity are used to localize functions, allowing for the extension of local properties to global ones. In particular, this is crucial in defining integration and differentiation on manifolds.
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