Proof: open sphere is an open set

[nuuskur]

Homework Statement

Prove that an open sphere in \mathbb{R}^m is an open set.

Homework Equations

The Attempt at a Solution

To show that an open sphere is an open set, any point inside the sphere has to be an interior point:
Let us have a sphere B(P_0, r), r > 0, where P_0 is the centerpoint and r is the radius of the sphere and also an arbitrary point P inside the sphere. Hence \exists\varepsilon>0\colon B(P,\varepsilon)\subset B(P_0, r) and as we assumed the B(P_0, r) to be open, then it does not contain its boundary.

The idea is to use the triangle inequality by taking another random point in a sphere around the already random point P in the large sphere: S\in B(P, \varepsilon). The distance between two points d(A,B) (Pythagorean)
d(S, P_0)\leq d(S,P) + d(P,P_0) <\varepsilon + d(P,P_0). This is a shaky part - how can I justify this inequality is always correct? I can visualize it in my head, but is it accurate?

Per that assumption I would fix \varepsilon\colon = r - d(P,P_0) > 0 since P is an interior point and its distance cannot be greater than or equal to the radius of the large sphere, therefore \varepsilon is always strictly positive and \forall S\in B(P, \varepsilon_1) I can state that the distance d(S,P_0) < \varepsilon + d(P,P_0) = r which would mean that the open sphere is an open set as all its points are interior points.

 

[PeroK]

The argument is essentially correct, but things are a bit mixed up and it doesn't flow. I suggest trying to tidy it up. I would start with something like this:

Let $B(P_0, r)$ be an open sphere in $\mathbb{R}^m$

Let $P \in B(P_0, r)$

$d(P_0, P) < r$ hence $\exists \epsilon > 0$ ...

 

[micromass]

A silly comment, but I don't like the term "open sphere". The terminology "open ball" is standard, because a sphere is usually the name given to the boundary of the ball, not something in the interior.

 

[nuuskur]

A silly comment, but I don't like the term "open sphere". The terminology "open ball" is standard, because a sphere is usually the name given to the boundary of the ball, not something in the interior.

I knew there was something off in the translation. I'll use ball to describe the set and sphere to describe its boundary, then.