A question on the compact subset of $[0,1]$

Question

Let $S=\{K \subseteq [0,1]: K \text{ is compact and uncountable } \}$. How to see that $|S|=\mathfrak c$?

Thanks for your help.

Answer 1

The compact subsets of $[0,1]$ are precisely the closed subsets of $[0,1]$, and there are $\mathfrak{c}$ many closed subsets. (In fact, there are precisely $\mathfrak{c}$ many Borel subsets.) Therefore $|S| \leq \mathfrak{c}$. Letting $K_0 = [0,\frac{1}{2}]$, it is easy to see that $K_0 \cup \{ a \} \in S$ for all $a \in (\frac 12,1]$, and we get $\mathfrak{c}$ many different sets in this fashion. Therefore $|S| \geq \mathfrak{c}$.