For a polynomial map $F:\mathbb{R}^n \rightarrow\mathbb{R}^n,$ let $F_{\delta}(\mathbf{x})$ denote the polynomial map obtained by perturbing each coefficient of $F$ by an amount less than $\delta.$ Let $V^R$ denote the regular solutions of the polynomial system $F(\mathbf{x})=0.$ That is, for each $x^* \in V^R,$ $F(\mathbf{x}^* )=0$ and the Jacobian matrix $F'(\mathbf{x}^*)$ is non-singular. Similarly, let $V^R_\delta$ denote the regular solutions of the polynomial system $F_\delta(\mathbf{x})=0.$ Then is it true that for any $\epsilon >0,$ one can find a small enough $\delta > 0$ so that

1) for each $x^* \in V^R,$ $\exists$ $y^* \in V^R_\delta$ such that $||x^* - y^*||_2 < \epsilon$

2) for each $y^* \in V^R_\delta, \exists$ $x^* \in V^R$ such that $||x^* - y^*||_2 <\epsilon$.