Let $X$ be an algebraic variety defined over $\mathbb C$. Let $x\in X$ be a fixed point. Let $O_x$ be the orbit of $x$ under rational equivalence, defined as $$ O_x:=\{y\in X: y\equiv_{rat} x \textrm{ in } X\},$$ where $y\equiv_{rat} x$ means rational equivalence.
It seems to be well-known that $O_x$ is a countable union of algebraic subsets. Could anyone give hints if it is true and why? Any comments or references are welcome.