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Laur
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Does every (continous and second countable) topological manifold have an Euclidean neighbourhood around each of its points whose closure equals the whole manifold?
Laur said:Does every (continous and second countable) topological manifold have an Euclidean neighbourhood around each of its points whose closure equals the whole manifold?
gel said:Yes!
Consider the set S of all neighbourhoods of a point P which are homeomorphic to the open ball. Try showing that this set has a maximal element (for this you just need to show that the union of a chain in S is again in S). Then that the closure of such a maximal element is the whole manifold.
I haven't worked through the details, but it seems like this argument should work.
wofsy said:Interesting. One can have a nested chain of open disks in a manifold whose union does not have the whole manifold as its boundary.
Take the height function on a torus standing on end on the the plane. From the bottom, f(x) < a, for a small enough is a a nested sequence of open disks that has a figure eight as its boundary and leaves out all of the torus above the first saddle point of f.
A topological manifold is a mathematical concept that describes a space that locally resembles Euclidean space, but may have a more complicated global structure. It is a type of topological space that is locally homeomorphic to Euclidean space.
A smooth manifold is a type of topological manifold that has the additional structure of a smooth atlas, which allows for a notion of differentiability. Topological manifolds do not have this additional structure, and therefore cannot be differentiated or integrated in the traditional sense.
Topological manifolds can represent a wide range of objects, including surfaces in 3D space, curves in higher-dimensional spaces, and abstract mathematical objects such as the set of real numbers or complex numbers.
Topological manifolds are important in mathematics because they provide a framework for studying and understanding geometric and topological properties of various objects. They also have applications in fields such as physics, engineering, and computer science.
Yes, there are many real-world examples of topological manifolds. For example, the surface of the Earth can be approximated as a topological manifold, with each point on the surface having a neighborhood that resembles a portion of a flat plane. Additionally, road maps, subway systems, and networks of blood vessels can all be modeled as topological manifolds.