Proving Equality of Interiors Under Homeomorphism

[Mandelbroth]

A friend gave me this to prove as part of an ongoing "game." I'm having a serious amount of difficulty with it, and I don't know what I need to do.

Homework Statement

"Prove the following:
If $U\subset\mathbb{R}^n$ is open, $A\subset U$ is homeomorphic to $S^{n-1}$, and $\varphi:U\to\mathbb{R}^n$ is a continuous bijection from $U$ to $\mathbb{R}^n$, then $\varphi(\operatorname{int} A)=\operatorname{int} \varphi(A)$."2. Relevant [strike]equations[/strike] lemma
My friend said that I "might" need to know that "if $B\subset\mathbb{R}^n$ is homeomorphic to $D^n=\left\{x\in\mathbb{R}^n : |x|\leq 1\right\}$, then $\mathbb{R}^n\setminus B$ is connected."

Here, the $|x|$ means $\displaystyle \sqrt{\sum_{i=1}^n x_i^2}$.

The Attempt at a Solution

I'm ashamed to say that I don't know where to start on this one. From the lemma that I "might" need, I know that $\mathbb{R}^n\setminus \operatorname{int} A$ is connected, but I don't know what this implies or how this gets me anywhere.

I'm really confused. I'd really appreciate any help anyone can give me.

 

[Office_Shredder]

I'm a bit confused, A is a codimension 1 submanifold of Rⁿ so has no interior.

 

[micromass]

Use invariance of domain.

 

[Mandelbroth]

Use invariance of domain.

So, by invariance of domain, we know that, since $\operatorname{int} A$ is open, then $\varphi(\operatorname{int} A)$ is also open, right? I also know that $\operatorname{int} \varphi(A)$ is open. This is at least something, but I don't know where this goes. Could you please give another hint?

 

[micromass]

Prove that if $\varphi$ is a homeomorphism, then for each set $B$ holds that $int \varphi(B) = \varphi(int(B))$.

 

[Mandelbroth]

Prove that if $\varphi$ is a homeomorphism, then for each set $B$ holds that $int \varphi(B) = \varphi(int(B))$.

I'm sorry. I knew that comes next because $\varphi$ being a homeomorphism follows from invariance of domain. I should have said that I don't know what property of a homeomorphism implies that equality. I think what I'm looking for can be stated in the form "[property] is conserved under homeomorphism," but I don't know.

Edit: Never mind. I figured it out. Invariance of domain implies homeomorphisms between subsets of $\mathbb{R}^n$ map interior points of one subset to interior points of the other. The result follows from this. Thank you for your patience.