# Proving an open ball is connected in a metric space X

1. Oct 9, 2013

### mahler1

The problem statement, all variables and given/known data.

Let $B(a,ε) (ε>0)$ in a metric space $(X,d)$. Decide whether this subset of $(X,d)$ is connected or not.

The attempt at a solution.
Well, I know open intervals in the real line are connected. I suppose that an open ball in a given metric space can be imagined as an open interval of a more general metric space instead of the real line; at least, that's the way I see it. So, by this analogy, I think that any open ball in a given metric space is always connected. My problem is I don't know how to prove it. I've tried it by the absurd:

So, suppose we can disconnect $B(a,ε)$. Then there exist $U$ and $V$, nonempty open subsets of $(X,d)$ such that
i)$B(a,ε)= U \cup V$
ii)$U \cap V=\emptyset$

How can I come to an absurd? By all the things above, I know that there exists $x \in B(a,ε)$ : $x \in U$ and $x \not\in V$. On the other hand, $U$ is an open set, so for some $δ>0$, $B(x,δ) \subset U$. That's all I got up to now. How can I continue to arrive to an absurd?

2. Oct 9, 2013

### gopher_p

Is it possible that the statement that you are trying to prove, that every open ball in every metric space is connected, is not true?

3. Oct 9, 2013

### Dick

The real line is too simple an example to make judgements based on it. Can't you think of a metric space that has disconnected open balls?

4. Oct 9, 2013

### mahler1

I got really mixed up trying to generalize the interval case. The counterexample I could think of was: consider X an infinite metric space with the discrete δ-distance. Pick any x in X and consider of the ball centered at x of radius 2. Then I can disconnect this ball by two open sets U and V consisting of union of open balls of radius 1 centered at any other point of the space. Is this correct? Can you give me another example to completely destroy my previous wrong assumption?

5. Oct 9, 2013

### mahler1

Yes, it's wrong, I got confused.

6. Oct 9, 2013

### Dick

Sure. Any subset of a metric space is a metric space, right? Take the subset of the reals X=(-infinity,-1/2]U[1/2,infinity) and think about the open ball around 0 of radius 1. There's also plenty of examples where X is connected and still has disconnected open balls. Can you think of one?

7. Oct 9, 2013

### Office_Shredder

Staff Emeritus
Dick, your example doesn't work because 0 isn't a point in the metric space to have an open ball around, but the open ball around -1/2 of radius 2 does work.

I second the think of a connected metric space which has disconnected open balls call. There are simple subsets of R2 that work.

8. Oct 9, 2013

### Dick

Good catch. Thank you!