Compact nowhere dense analytic closed set

Question

For a compact nowhere dense analytic closed set, why is that a finite set?

Can we get this set is discrete, so that it's finite?

Analytic sets are locally zero sets of holomorphic functions, which means to an analytic set $A$, to any $x\in A$, there exists a neighborhood $U_x$, s.t. $A\cap U_x=\{ y\in U_x| f_1(y)=...=f_n(y)=0\}$ for some holomorphic functions $\{f_i(x)\}$.

Answer 1

If you are using the standard definition of analytic sets as in the link below then the statement it false: the Cantor set is a Borel set in $[0,1]$, hence analytic. It is nowhere dense but not finite.

https://en.wikipedia.org/wiki/Analytic_set