1. ## Isolated point

Let a be an element of A. Prove that A is an isolated point of A iff there exists an epsilon neighborhood V(a) such that V(a)intersectA={a}

A point is an isolated point if it is not a limit point.
Let a be an element of A.
Let be an isolated point. We want to show V(a)intersectA={a}.
Since a is not a limit point, we say x=liman satisfying an=x

2. Originally Posted by kathrynmath
Let a be an element of A. Prove that A is an isolated point of A iff there exists an epsilon neighborhood V(a) such that V(a)intersectA={a}
A point is an isolated point if it is not a limit point.
To say that $x\notin A^{\prime}$, x is not a limit point of A, then there is a neighborhood $V(x)$ that does not any point of $A\setminus \{x\}$.
Therefore, if $x\in A$ then $V(x)\cap A=\{x\}$, so it is isolated.

3. I don't get where A' comes from?