# Connected quotient under conjugate action

I am working on a problem in hyperplane arrangement theory and I would like to apply the Randell Isotopy Theorem in order to characterize the diffeomorphism type of some specific class of arrangements.

In particular, I am interested in studying the irreducibility and the connectedness of the realization space of a prescribed matroid over the complex field.

I formulate my question in a more general setting. Here is the problem.

Set up: Let $p_{1},\ldots,p_{m}\in\mathbb{R}[x_{1},\ldots,x_{d}]$ and let us consider the space $$X=\{(z_{1},\ldots,z_{d})\in\mathbb{C}^{d}\mid p_{j}(z_{1},\ldots,z_{d})=0\text{ for }j=1,\ldots,m\}$$ Let $Y$ be the quotient space $Y=X/\sim$, where $\sim$ denotes the conjugate action. To be more precise,

$$(z_{1},\ldots,z_{d})\sim(z_{1}',\ldots,z_{d}')$$ if and only if $(z_{1},\ldots,z_{d})=(z_{1}',\ldots,z_{d}')$ or $(z_{1},\ldots,z_{d})=(\overline{z}_{1}',\ldots,\overline{z}_{d}')$.

Let us assume $X$ is endowed with the standard topology (not the Zariski one) and let us assume $Y$ is endowed with the quotien topology of $X$. Finally, let us assume $Y$ is connected.

Question: Given $(z_{1},\ldots,z_{d})$ and $(z_{1}',\ldots,z_{d}')$ elements of $X$ is that true that either $(z_{1}',\ldots,z_{d}')$ or $(\overline{z}_{1}',\ldots,\overline{z}_{d}')$ is in the same connected component of $(z_{1},\ldots,z_{d})$?