A basic question in the definition of limit point

Question

For any subset of $R$ with the usual distance metric, any point inside it is a limit point. Only when the set is discrete there may be a point inside it which is not a limit point. Is this correct ?

Another question, according to this definition in the $(0, 1)$ open interval any point inside it is a limit point. What is the meaning of, for example, $\frac{1}{2}$ is a limit point of $(0,1)$. I understand the meaning that $0$ and $1$ is a limit point of $(0,1)$ in the sense that each of them approximates the one side of $(0,1)$.

BTW, I am new to analysis

Answer 1

A limit point $p$ of set $S$ in a metric space is such that for any $\epsilon\in\Bbb{R}$, the ball $(B(p,\epsilon)\setminus \{p\})\bigcap S\neq\emptyset$. This means that every such ball $B(p,\epsilon)$ contains points from $S$ aside from $p$ (if $p$ is a part of $S$ too).

Now think about $(0,1)$ and its limit point $\frac{1}{2}$. You will get your answer.

In topoogy in general, a limit point need not just approximate one side of an interval. Spending some time with this new definition of a limit point helped me learn the suject better.

Answer 2

A set need not be discrete to contain points that are not limit points of the set. Let $A=[0,1]\cup\{2\}$; then $2$ is not a limit point of $A$, but $A$ is very far from being discrete, since every other point of $A$ is a limit point of $A$.

A point $x$ is a limit point of a set $A$ if every open interval $(u,v)$ containing $x$ contains at least one point of $A$ different from $x$. More technically, $x$ is a limit point of $A$ if for each $u,v\in\Bbb R$ such that $u<x<v$, we have $(u,v)\cap(A\setminus\{x\})\ne\varnothing$. If $A=(0,1)$ and $x=\frac12$, this is clearly the case: if $u<\frac12<v$, $(u,v)\cap(0,1)$ certainly contains numbers other than $\frac12$. Indeed, let $$y=\max\left\{\frac14,\frac12\left(u+\frac12\right)\right\}\;;$$ it’s a good exercise for you to show that $y\in(u,v)\cap(0,1)$. (HINT: $\frac12\left(u+\frac12\right)$ is the midpoint of the segment from $u$ up to $\frac12$.)