# Interior Points are Open

1. Jan 16, 2013

### pokemonsters

1. The problem statement, all variables and given/known data
For S $\subset$ Rn, prove that S° is open.

2. Relevant equations
S° are all interior points of S.

3. The attempt at a solution
My class has only learned how to use balls to solve these types of problems (no metric spaces). So I need to choose an ε > 0 so that Bε(x) $\subset$ S°, where x is any arbitrary point in S°. To show this is true, let y $\subset$ Bε(x) be arbitrary. (then I don't know how to progress further...how do I show that the neighbourhood contains only points in S°?)

2. Jan 16, 2013

### jbunniii

Start by choosing an arbitrary $x \in S^o$. By definition this is an interior point of $S$, so there exists $\epsilon > 0$ such that $B_\epsilon(x) \subset S$. Now, if you can show that every point $y \in B_\epsilon(x)$ is an interior point of $S$ then you're done. To do this, it certainly suffices to show that you can fit a smaller ball $B_\delta(y)$ around $y$ which is entirely contained within $B_\epsilon(x)$, because then you will have $y \in B_\delta(y) \subset B_\epsilon(x) \subset S$. Try drawing a picture to see how to define $\delta$, the radius of the smaller ball.

3. Jan 17, 2013

### pokemonsters

Thank you, I did not notice that you can put another ball inside the ball to make the proof work.

Using this, I was able to make a series of inequalities using the triangle inequality, and managed to prove that $y \in B_\delta(y) \subset B_\epsilon(x) \subset S$.