A Real Analytic Variety $N$ of Real Dimension $n$ of a Complex Manifold $M$ of Complex Dimension $m$

Question

Let $M$ be a Complex Manifold of complex dimension $m$.

A Real Analytic Variety of $M$ is a closed subset $N \subseteq M$ such that for any point $p\in N$ there exists an open neighbourhood $p\in U \subseteq M$ such that $N \cap U$ is the zero set for finitely many real analytic functions $f_1,f_2,..,f_k: U \longrightarrow \mathbb{R}$, i.e., $$N \cap U=\{z\in U: f_1(z) = f_2(z) = ... = f_k(z) = 0 \}$$

The questions are:

1. Is the previous definition correct?
2. When can we say that $N$ has real dimension $n=2m-1$?
3. When can we call $N$ as a Singular real analytic variety?