Q:

Isolated points over specific intervals

I am having some trouble understanding isolated points.  I am using the textbook "Real Analysis" by Manfred Stoll.
Isolated points are defined as "a point p in set E that is not a limit of E."  Limit point:  A point p in the real numbers is a limit point of E if every epsilon-neighborhood N(p) of p contains a point q in set E with q does not equal p."
I am trying to understand these definitions a little better by looking at intervals and sets.
I believe the limit points of (a,b), (a,b], and [a,b] are all [a,b] by the definition.  Does this mean they have no limit points?  I also looked at some more complicated sets the Rational numbers Q.  Namely, [0,1] intersect Q.  I believe the limit points are [0,1] intersect Q since every point in Q is a limit point.  Does this mean there are no isolated points?  Any help with explanation is greatly appreciated!

A:

You are mostly right. The limit points of all the sets $(a,b)$, $(a,b]$, and $[a,b]$ are all given by the set $[a,b]$. Every point in the set is a limit point, and these intervals contain no isolated points. The set
$$P=[a,b]\bigcap\mathbb{Q}$$
has limit points $[0,1]$. This may seem odd that all of the irrationals between $0$ and $1$ are limit points of $P$, but it is true because any for number $x\in[0,1]$ and any $\epsilon>0$, the $\epsilon$ neighborhood surrounding $x$ must contain a rational number. So $P$ has no isolated points.

