A basic question on limit point of an infinite subset Suppose in a metric space $X$ for a given $\delta >0$ we can find an infinite subset of $X$ in which the distance between any two elements is $\geq \delta$. Then can we say anything about whether the set has a limit point or not ?  

Let $A$ be such a set. If $x\in A$, then $N(x,\delta)\cap A=\{x\}$, so $x$ is not a limit point of $A$. If $x\in X\setminus A$, there are two possibilities. If there is an $a\in A$ such that $d(x,a)<\delta$, let $\epsilon=d(x,a)$; then $N(x,\epsilon)\cap A=\varnothing$, so $x$ is not a limit point of $A$. And if there is no such $a\in A$, then $N(x,\delta)\cap A=\varnothing$, and again $x$ is not a limit point of $A$.
We can also conclude that $X$ is not compact: $A$ is closed, so $$\mathscr{U}=\{N(x,\delta):x\in A\}\cup\{X\setminus A\}$$ is an open cover of $X$ with no finite subcover, since if $x\in A$, $N(x,\delta)$ is the only member of $\mathscr{U}$ containing $x$.
