limit vs isolated point

Definition(from Definition 2.18, Principles of Mathematics Analysis by Walter Rudin):

Whenever Nr(p)∩E≠ф, p is called a limit point of E.

If every limit point of E is a point in E, then E is closed.

If p is in E but not a limit point of E then p is called an isolated point of E.

*def:Nr(p)=every neighborhood of p with radius r

i think i've already studied until gila le...

it makes me think of p is just like myself, whenever any1 in my neighborhood is in the group(E), i always have a "relationship" with E(called "limit point" but i called it "boundary")...unless i'm isolated in E but no neighborhood in E, or never be a point of E...