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

[ehrenfest]

Homework Statement

This is a topology.

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

If I want to show that $$f|_{A}$$, 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

 

[morphism]

Yes. That will prove that f|_A is a homeomorphism between A and f(A).

 

[ehrenfest]

Thanks. That helps!