zeroes of holomorphic function I know that zeroes of holomorphic functions are isolated,and I know that if a holomorphic function has zero set whic has a limit point then it is identically zero function,i know a holomorphic function can have countable zero set, does there exixt a holomorphic function which is not identically zero, and has uncountable number of zeroes?

A holomorphic function on a connected open set that is not identically zero cannot have uncountably many zeros.  Open subsets of $\mathbb{C}$ are $\sigma$-compact, so if $G$ is the domain, then there is a sequence $K_1,K_2,\ldots$ of compact subsets of $G$ such that $G=K_1\cup K_2\cup\cdots$.  (It is not hard to construct $K_n$; e.g., if $G$ is the whole plane, you can take $K_n=\{z:|z|\leq n\}$.  Otherwise you can take $\{z\in G:|z|\leq n\text{ and }d(z,\partial G)\geq \frac{1}{n}\}$.) An uncountable subset of $G$ must have uncountable intersection with one of the $K_n$s, because a countable union of countable sets is countable.  An infinite subset of $K_n$ has a limit point in $K_n$ by compactness.  The rest follows from the result you mentioned that a holomorphic function that is not identically zero cannot have a limit point of its zero set in the connected open set on which it is defined.
