No, let $X$ be the of those irrationals in $x\in (0,1)$ with binary expansion
$$x=0.x_1x_2\dots$$
such that if we use a bijection $\mathbb N^2\to\mathbb N$ to represent $x$ as infinitely many reals $x^{(1)}, x^{(2)},\dots$ then
$$\lim_{n\to\infty} 1_{\mathbb Q}(x^{(n)})$$
exists.

Here $1_{\mathbb Q}(y)=1$ if $y\in\mathbb Q$ and 0 otherwise.