I have been self studying real analysis and came across a question regarding isolated vs interior points. I am reading that an isolated point p of a set S is an element of S such that p not a limit point of S; however, a limit point of S is a point such that any neighborhood of p contains a point q not equal to p in S. My question is, wouldn't any point in a set have a neighborhood that contains a different point in that set, and therefore be a limit point?
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For limit points the requirement is that every neighborhood of that point has a point of set different from that point.
As the result every neighborhood will have infinitely many points of the set different from that point. Another way of saying it is the limit point is actually the limit of a sequence of distinct points of the set converging to that point.
No. Or, more accurately, yes, but the thing you wrote after "my question is" is not the same as the definition: for a point $p$ to be a limit point of a set $S$, every neighbourhood of $p$ must contain some element of $S \setminus \{p\}$.
It is then easy to see that not all points have this property: for example, for any $p \in \mathbb{Z}$, there is no point of $\mathbb{Z}$ in the neighbourhood $(p - 1, p + 1)$ of $p$.