Boundary and homeomorphism

In summary, the statement is true if and only if the following conditions are met:-bA is the boundary of the subset A of X-F(bA) is the boundary of the subset F(A) of Y
  • #1
aleazk
Gold Member
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Hi, I need to know if the following statement is false or true. Given two topological spaces, X and Y, and an homeomorphism, F, between them, if bA is the boundary of the subset A of X, this implies that F(bA) is the boundary of the subset F(A) of Y?
 
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  • #2
What did you try already?? What is the definition of the boundary??
 
  • #3
sorry if this is a silly question, I'm new in topology. bA=CA-IA, where CA is the closure and IA is the interior.
 
  • #4
aleazk said:
sorry if this is a silly question, I'm new in topology. bA=CA-IA, where CA is the closure and IA is the interior.

Indeed, so what happens if you take

[tex]f(CA-IA)[/tex]

can you show that this equals [itex]Cf(A)-If(A)[/itex]??

Simplified, can you show that

[tex]Cf(A)=f(CA)~\text{and}~If(A)=f(IA)[/tex]
 
  • #5
aleazk said:
sorry if this is a silly question, I'm new in topology. bA=CA-IA, where CA is the closure and IA is the interior.

That is equivalent to saying that point p is in the boundary of A if and only if any open set containing p contains points in A and points not in A. And, of course, homeomorphisms map open sets to open sets.
 
  • #6
HallsofIvy said:
That is equivalent to saying that point p is in the boundary of A if and only if any open set containing p contains points in A and points not in A. And, of course, homeomorphisms map open sets to open sets.

So, using the property AcB then F[A]cF of maps, the open sets that contain F(p) will also contain points inside and outside of F[A], and then F(p) is in the boundary of F[A] if p is in the boundary of A?
 
  • #7
aleazk said:
So, using the property AcB then F[A]cF of maps, the open sets that contain F(p) will also contain points inside and outside of F[A], and then F(p) is in the boundary of F[A] if p is in the boundary of A?


Indeed! You may want to rigorize that however. For example, why will open sets containing F(p) also contain points in and out F(A)?
 
  • #8
micromass said:
Indeed, so what happens if you take

[tex]f(CA-IA)[/tex]

can you show that this equals [itex]Cf(A)-If(A)[/itex]??

Simplified, can you show that

[tex]Cf(A)=f(CA)~\text{and}~If(A)=f(IA)[/tex]
Hi, I suppose yes. int A is the largest open set contained in A, so int A c A and then F[int A] c F[A]. If F is an homeomorphism, then F^-1 is continuous, which implies that F[int A] is open. Then, because the last relation, F[int A] c F[A], F[int A] c int F[A]. Now, int F[A] c F[A], so F^-1[int F[A]] c A. Because F is continuous, then F^-1[int F[A]] is open. Using the last relation, F^-1[int F[A]] c A, it follows that F^-1[int F[A]] c int A, which is equivalent to int F[A] c F[int A]. Thus, using the previous result, F[int A] c int F[A], it follows that F[int A] = int F[A]. CA is the smallest closed set containing A, so F[A] c CF[A] and then A c F^-1[CF[A]]. F^-1[CF[A]] is closed because CF[A] is closed and F is continuous. Then CA c F^-1[CF[A]], which is equivalent to F[CA] c CF[A]. Using similar arguments, but invoking the continuity of F^-1, it follows that CF[A] c F[CA]. So, F[CA] = CF[A].
 
  • #9
aleazk said:
Hi, I suppose yes. int A is the largest open set contained in A, so int A c A and then F[int A] c F[A]. If F is an homeomorphism, then F^-1 is continuous, which implies that F[int A] is open. Then, because the last relation, F[int A] c F[A], F[int A] c int F[A]. Now, int F[A] c F[A], so F^-1[int F[A]] c A. Because F is continuous, then F^-1[int F[A]] is open. Using the last relation, F^-1[int F[A]] c A, it follows that F^-1[int F[A]] c int A, which is equivalent to int F[A] c F[int A]. Thus, using the previous result, F[int A] c int F[A], it follows that F[int A] = int F[A]. CA is the smallest closed set containing A, so F[A] c CF[A] and then A c F^-1[CF[A]]. F^-1[CF[A]] is closed because CF[A] is closed and F is continuous. Then CA c F^-1[CF[A]], which is equivalent to F[CA] c CF[A]. Using similar arguments, but invoking the continuity of F^-1, it follows that CF[A] c F[CA]. So, F[CA] = CF[A].

Seems ok! :smile: Nicely done!
 
  • #10
micromass said:
Seems ok! :smile: Nicely done!

Thanks. I needed the result to convince myself of a claim that I read below proposition 4.5.1 of Hawking and Ellis. The proposition says that in a convex normal neighborhood Np of point p, the points in Np that can be reached by timelike curves diverging from p are those of the form Q=exp\p(V), where V is timelike. But then, below the proposition, he says: in other words, the null geodesics that diverge from p form the boundary of the region in Np that can be reached by timelike curves diverging from p. :confused: why?
The zone exp\p(V), where V is timelike, is generated by the timelike geodesics that diverge from p, and the zone exp\p(V), where V is null, is generated by the null geodesics that diverge from p, all this by definition of the exponential map at p. In the tangent space to p, Tp, the null vectors form the boundary of the zone where the timelike vectors lie. Thus, because exp\p is a diffeomorphism (and then an homeomorphism) at Np, using the result we discused in this tread, the null geodesics that diverge from p really form the boundary of the zone generated by the timelike geodesics that diverge from p. At least I think so :redface:
 

1. What is the definition of a boundary in mathematics?

A boundary in mathematics is a set of points that separates one set from another. It is the set of points that are not contained within the set, but are in the immediate vicinity of the set. In other words, it is the edge or perimeter of a set.

2. How is the boundary of a set related to its interior and exterior?

The boundary of a set is the boundary between its interior and exterior. It is the set of points that are in the immediate vicinity of the set, but are not contained within it. The interior of a set is the set of points that are completely contained within the set, while the exterior is the set of points that are completely outside of the set.

3. What is a homeomorphism and how does it relate to boundaries?

A homeomorphism is a continuous function between two topological spaces that has a continuous inverse function. It preserves the topological structure of the space, meaning that the points that are close together in one space will still be close together in the other space. Homeomorphisms are often used to identify boundaries between sets, as they preserve the boundary points and their neighborhoods.

4. Can a set have more than one boundary?

No, a set can only have one boundary. This is because the boundary is defined as the set of points that are in the immediate vicinity of the set, but not contained within it. If there were multiple boundaries, there would be overlap between the sets, and they would no longer be distinct sets.

5. How do boundaries play a role in the study of topology and geometry?

In topology and geometry, boundaries are important for understanding the structure of a set and its relationship to other sets. They help define the interior and exterior of a set, and can also be used to identify homeomorphisms between sets. Boundaries are also important in the study of manifolds, which are topological spaces that locally resemble Euclidean spaces.

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