If $A$ is a subset of a compact space such that every point of $A$ is an isolated point, is $A$ necessarily finite?

Question

Let $A$ be a subset of a compact topological space such that every point of $A$ is an isolated point of $A$. Is $A$ necessarily finite?

Answer 1

No. Take for instance the compact space $[0,1]$, and let $A=\{1/n:n\in\Bbb N_{>0}\}$.

Answer 2

No, it can be very large. But if $A$ is also closed, it is compact and then it must be finite.

E.g. $[0,1]^{\mathbb{R}}$ is compact in the product topology, but the set $A$ of all functions $\{f_t: \mathbb{R} \to [0,1], t \in \mathbb{R}\}$ defined by $f_t(x) = 0$ if $x \neq t$ and $f_t(t)=1$, consists only of isolated points, as $\pi_t^{-1}[(\frac12,\frac32)] \cap A = \{f_t\}$ e.g. So $A$ can be very large, but non-closed.