Accumulation points of uncountable sets

Given any uncountable subset $S$ of the unit interval. Then $S$ clearly has an accumulation point and indeed uncountably many (which might also be a nice exercise). So my question is: Is there an accumulation point, that again lies in $S$?

Solution 1:

Yes, every uncountable subset of $\mathbb{R}$ contains at least one of its accumulation points.

Suppose otherwise, that $S \subseteq \mathbb{R}$ is uncountable and contains none of its accumulation points. Then for each $x \in S$ there is an $\epsilon_x > 0$ such that $( x - \epsilon_x , x + \epsilon_x ) \cap S = \{ x \}$. Then there is an $n \in \mathbb{N}$ such that $S^\prime = \{ x \in S : \epsilon_x > \frac 1n \}$ is uncountable. Consider the family $$\{ ( x - \tfrac {1}{2n} , x + \tfrac{1}{2n} ) : x \in S^\prime \}.$$ It can be shown that this is an uncountable family of pairwise disjoint open subsets of $\mathbb{R}$ which contradicts that the countable set $\mathbb{Q}$ is a dense subset of $\mathbb{R}$.

Solution 2:

Yes; you can even take it to be a complete accumulation point.

Let $\mathscr{B}$ be any countable base for the topology of $\Bbb R$, e.g., the set of open intervals with rational endpoints. Let $$C=\{x\in S:\exists B\in\mathscr{B}(x\in B\text{ and }B\cap S\text{ is countable})\}$$ be the set of points of $S$ having an open nbhd containing only countably many points of $S$. For each $x\in C$ let $B_x\in\mathscr{B}$ be such that $x\in B_x$ and $B\cap S$ is countable, and let $\mathscr{B}_0=\{B_x:x\in C\}$. Then $\mathscr{B}_0$ is countable, so $$C=\bigcup_{B\in\mathscr{B}_0}(B\cap S)$$ is countable as well, and every point of $S\setminus C$ is a complete accumulation point of $S$.

Solution 3:

If any accumulation point of $S$ does not lie in $S$, then $S$ is discrete; so for any $s \in S$, there is an interval $I_s$ of length $\ell_s>0$ such that $S \cap I_s = \{s\}$. Moreover, $\displaystyle \sum\limits_{s \in S} \ell_s \leq \ell g([0,1])=1$. So $S$ is at most countable. Indeed, since $\displaystyle \sum\limits_{s \in S} \ell_s$ is finite, then $\{s \in S : \ell_s > 1/n \}$ is finite, hence $\displaystyle S= \{s \in S : \ell_s >0 \}=\bigcup\limits_{n >0} \{s \in S : \ell_s > 1/n \}$ is at most countable.