Is this injective projection map open?

Question

Let X be a locally closed algebraic sub variety of $\mathbb C^{m+n}$. Let $p: \mathbb C^{m+n}\to \mathbb C^{m}$ be the projection map. Suppose that the restriction of p to X is injective. Is the map $p: X \to p(X)$ open in Euclidean topology? Note that p(X) is Zariski constructible in $\mathbb C^{m}$.