Constructing a sequence in a manifold

In summary, the conversation discusses proving that a submanifold S is embedded. It is assumed that S is not embedded and a point p is chosen that is not contained in any slice chart. By considering a smooth function on S that is supported in a neighborhood U of p and equal to 1 at p, a contradiction is reached if a sequence x_n in S-U converges to p. This sequence can be constructed using the first countability of manifolds.
  • #1
JYM
14
0
Given S is a submanifold of M such that every smooth function on S can be extended to a smooth function to a neighborhood W of S in M. I want to show that S is embedded submanifold.
My attempt: Suppose S is not embedded. Then there is a point p that is not contained in any slice chart. Since a submanifold is locally embedded, let U be a neighborhood of p that is embedded. Consider a smooth function on S that is supported in U and equal to 1 at p. If there is a sequence x_n in S-U that converges to p, then since f can be extended to a smooth function F on W but then 0=F(x_n) converges to F(p)=1, which is contradiction. My difficulty is to justify such a sequence exists. Please provide me your help.
 
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  • #2
No. We can construct such a sequence. Now I get the idea; the result follows from first countablity of manifolds ( as second countable is first countable).
 

1. What is a manifold?

A manifold is a mathematical concept used in geometry and topology to describe a space that is locally similar to Euclidean space. In other words, a manifold is a space that can be smoothly and continuously mapped onto a portion of a flat space, such as a plane or a sphere.

2. How is a sequence constructed in a manifold?

A sequence in a manifold is constructed by choosing a starting point, or initial value, and then using a series of steps or transformations to move from one point to the next. These steps must follow the rules and properties of the manifold in order to maintain the smoothness and continuity of the sequence.

3. What are some examples of manifolds?

Some common examples of manifolds include spheres, tori, and projective spaces. In physics, spacetime is also considered a four-dimensional manifold. In general, any space that can be defined by a set of coordinates and has a smooth structure can be considered a manifold.

4. What is the importance of constructing a sequence in a manifold?

Constructing a sequence in a manifold allows us to study and understand the behavior of a system or process that is defined on that manifold. It also allows us to make predictions and draw conclusions about the behavior of the system at different points in the manifold.

5. Can a sequence in a manifold have an infinite number of points?

Yes, a sequence in a manifold can have an infinite number of points. This is because manifolds can be infinite in extent, such as the real line, or they can be discrete, meaning they have a countable number of points. In both cases, a sequence can be constructed by choosing a starting point and following a set of steps indefinitely.

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