- #1

mahler1

- 222

- 0

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?