cianfa72's latest activity
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cianfa72 reacted to fresh_42's post in the thread I About the definition of topological manifold using closed sets with
Like.
The key point is that you can choose whatever small neighborhood around a point and still have the same behavior. A closed set might... -
cianfa72 replied to the thread I About the definition of topological manifold using closed sets.Ah yes, well spotted. The same issue doesn't occur when using open sets to define neighborhoods. So, by using closed sets, we no longer...
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cianfa72 reacted to martinbn's post in the thread I About the definition of topological manifold using closed sets with
Informative.
Let's take our topological space to be the plane ##M=\mathbb R^2##. How do you make it a topological manifold using closed sets? Every... -
cianfa72 replied to the thread I About the definition of topological manifold using closed sets.Use this -- Definition using closed sets. The definition of chart ##(U, \phi)## is based on closed sets using homemorphism on the image...
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cianfa72 replied to the thread I About the definition of topological manifold using closed sets.Define the topologies involved by using axioms for closed sets (i.e. arbitrary intersections and finite unions instead of vice versa)...
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cianfa72 replied to the thread I About the definition of topological manifold using closed sets.Sorry, can you better explain what is the trouble with singletons set ?
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cianfa72 replied to the thread I About the definition of topological manifold using closed sets.Nothing. I asked it just to better understand how topological vs differential elements enter the definition of manifold (topological vs...
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cianfa72 posted the thread I About the definition of topological manifold using closed sets in Topology and Analysis.We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in...
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cianfa72 replied to the thread I About the maximal extension of local charts on a manifold.In the first case (rational ratio) vector fields ##(1,2)## and ##(3,4)## are smooth, pointwise linearly independent, commute and their...
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cianfa72 replied to the thread I About the maximal extension of local charts on a manifold.Ah ok, ##(1,2)## for instance is a smooth vector field that has constant components in coordinate basis ## \{\partial / \partial...
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cianfa72 replied to the thread I About the maximal extension of local charts on a manifold.Sorry, not sure to understand which are the two your linearly independent vector fields ##(\omega_1,\omega_2)## and...
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cianfa72 replied to the thread I About the maximal extension of local charts on a manifold.Yes. Better, I'd say since a smooth manifold is first and foremost a topological manifold, we can cover it iff, as topological manifold...
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cianfa72 reacted to wrobel's post in the thread I About the maximal extension of local charts on a manifold with Like.
you can cover a manifold with a single chart if only this manifold is diffeomorphic to an open subset of ##\mathbb{R}^m##. I do not...
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cianfa72 posted the thread I About the maximal extension of local charts on a manifold in Differential Geometry.I'm a bit confused by the conditions on the existence of coordinate basis given by Frobenius's theorem. Namely, let's take a...
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cianfa72 replied to the thread I About the notion of non-standard inertial frame.Ah yes, you're right (I edited my post #20). Coming back to what @Ibix said in post #10, I've not a clear understanding why a non...