Let $X = (X_1,..,X_n)$ and $I$ be an ideal in the ring $Q[X]$ ( polynomials with rational coefficients) and $V_{\mathbb{R}}(I)$ be the real variety of $I$. Let $x = (x_1,...,x_n)$ be a point in this variety and suppose the coordinates $x_1,...,x_m$ ($m < n$) is algebraically independent over $Q$. Let $P_m(V_{\mathbb{R}}(I))$ be the projection of real variety $V_{\mathbb{R}}(I)$ onto coordinates $X_1,...,X_m$.
Is this true that there exists a neighbourhood for each coordinate such that we can perturb each coordinate $x_i$ and it remains in $P_m(V_{\mathbb{R}}(I))$? In other words, does the point $(x_1,...,x_m)$ lie in the interior of $P_m(V_{\mathbb{R}}(I))$?