Proving Homeomorphisms of Restricted Functions: Do These Steps Ensure Success?

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In summary, to show that f|_{A} is a homeomorphism, we need to show that the inverse of U in the subspace topology of f(A) (w.r.t. Y) is open in the subspace topology of A (w.r.t. X), and that U in the subspace topology of A (w.r.t X) is open in the subspace topology of f(A) (w.r.t. Y). Once these conditions are met, f|_{A} is a homeomorphism between A and f(A).
  • #1
ehrenfest
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Homework Statement


This is a topology.

Let f: X -> Y be a bijective function.

If I want to show that [tex] f|_{A}[/tex], where A is a set in X, is a homeomorphism, I need to show that:

1)If U is open in the subspace topology of f(A) (w.r.t. Y), then f^{-1}(U) is open in the subspace topology on A (w.r.t X)

2)If U is open in the subspace topology of A (w.r.t X), then f(U) is open in the subspace topology on f(A) (w.r.t. Y)

and then I am done, correct?

Homework Equations


The Attempt at a Solution

 
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  • #2
Yes. That will prove that f|A is a homeomorphism between A and f(A).
 
  • #3
Thanks. That helps!
 

1. What is the definition of restriction of a function?

The restriction of a function is a process in which the domain of the original function is limited or restricted to a smaller subset of values. This results in a new function with a smaller domain but the same range as the original function.

2. Why is restriction of a function important?

Restriction of a function allows us to focus on a specific part of a function and analyze its behavior. It also helps in simplifying complex functions and making them more manageable to work with.

3. How is the restriction of a function represented mathematically?

The restriction of a function f(x) to a set A is represented as f|A or f|A. This indicates that the function is being restricted to the values in the set A.

4. Can the restriction of a function change its range?

No, the range of a restricted function will always be the same as the range of the original function. The restriction only affects the domain of the function.

5. Are there any limitations on the type of functions that can be restricted?

No, any type of function, whether it is linear, quadratic, exponential, etc., can be restricted as long as the new domain is a subset of the original domain. However, some functions may not have a restricted form that is simpler to work with.

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