Prove that topological manifold homeomorphic to Euclidean subspace

In summary, the conversation discusses the proof that every topological manifold can be homeomorphic to a subspace of n-dimensional Euclidean space. The solution involves constructing a subspace and a corresponding homeomorphic mapping for each point, possibly using a partition of unity to extend the local maps to a global map.
  • #1
sunjin09
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0

Homework Statement


Show that every topological manifold is homeomorphic to some subspace of E^n, i.e., n-dimensional Euclidean space.


Homework Equations


A topological manifold is a Hausdorff space that are locally Euclidean, i.e., there's an n such that for each x, there's a neighborhood N(x) homeomorphic to E^n.


The Attempt at a Solution


This must be well known, but I have no idea how to start. How would I construct the E^n subspace and the corresponding homeomorphic mapping? By definition, there's a different mapping for each point. Thank you for your help.
 
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  • #2
perhaps a partition of unity? you need to find a way of extending the local maps to generate a global map inheriting the properties of the local maps by restriction.
 

1. What is a topological manifold?

A topological manifold is a topological space that locally resembles Euclidean space. It is a type of mathematical object that is often used in the study of geometry and topology.

2. What is a homeomorphism?

A homeomorphism is a continuous function between two topological spaces that has an inverse that is also continuous. In simpler terms, it is a mapping between two spaces that preserves their topological properties.

3. How do you prove that a topological manifold is homeomorphic to a Euclidean subspace?

To prove that a topological manifold is homeomorphic to a Euclidean subspace, you need to construct a continuous function between the two spaces that satisfies the definition of a homeomorphism. This can be done by showing that the two spaces have the same topological properties, such as connectedness, compactness, and Hausdorffness.

4. What are some examples of topological manifolds that are homeomorphic to Euclidean subspaces?

Some examples include the 2-dimensional sphere (a 2-dimensional surface that is homeomorphic to a 2-dimensional Euclidean plane), the 3-dimensional torus (a 3-dimensional surface that is homeomorphic to a 3-dimensional Euclidean space), and the 4-dimensional sphere (a 4-dimensional surface that is homeomorphic to a 4-dimensional Euclidean space).

5. How is the concept of homeomorphism useful in mathematics?

Homeomorphisms are useful in mathematics because they allow us to study topological spaces by mapping them to simpler spaces that we already understand, such as Euclidean spaces. They also help us to identify and classify different types of spaces based on their topological properties.

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