Show that if $E \subseteq X$ is connected and has at least two points, then $E$ has no isolated points. Thus a connected set with at least two points must contain infinitely many points.
I understand that if $E$ has no isolated points, it must be infinite, because no "ball" around the point will be such that it doesn't contain any element of E, but how do I write this mathematically?
I also understand that if $E$ is connected and has at least two points, it must contain infinitely many points because it is connected and so you must be able to have a "ball" around either point containing at least one element of $E$ by the definition of connectedness. I'm having trouble writing this mathematically and I feel I'm missing some intuition.
Here, $X$ is a nonempty set equipped with a metric $d$.
Best Answer
Hint Consider $a,b\in E$ consider $f:E\to\mathbb{R}$ defined by $f(x)=d(x,a)$ which is a continuous function , $f(a)=0,f(b)>0$ , $f(E)$ is connected and it is an interval.