Is There a Connected Union of Open Balls in a Metric Space?

[e(ho0n3]

Let X be a connected metric space, let a, b be distinct points of X and let r > 0. Is there a collection {B_i} of finitely many open balls of radius r such that their union is connected and contains a and b.

I was trying to prove this by contradiction, but couldn't derive a contradiction. I can't think of any counterexamples either. Does anyone know whether this statements is true or false?

 

[Hurkyl]

Hrm. Well, let's say that a and b are r-ball-connected if such a collection exists.

A common argument regarding variants of connectedness is to consider something like

A(a,r) = the set of all points that are r-ball connected to a.

Now, A(a,r) is clearly an open set. Is it closed? (I'm assuming you know the ramifications of a set being both open and closed)

 

[e(ho0n3]

Great tip. Thanks for clearing that up.