Must a subset of the real line that is comprised entirely of condensation points be a Baire space?

Question

Let $X$ be a subset of the real line in which every point is a condensation point. Is $X$ a Baire space?

Answer 1

Not necessarily. For $0<r<1/2$, let $C_r$ be the Cantor set obtained by removing $(r, 1-r)$ segments of intervals as follows, $$C_r=([0, r]\cup [1-r, 1])\bigcap ([0, r^2]\cup [r-r^2, r]\cup [1-r, 1-r+r^2]\cup[1-r^2, 1])\bigcap\ldots$$

Each $C_r$ is closed and every point of $C_r$ is a condensation point. Now consider $$X=\bigcup\limits_{q\in (0, 1/2)\cap \mathbb{Q}} C_q.$$

Every point of $X$ is a condensation point and each $C_q$ has empty interior in $X$. Indeed, fix $a\in C_q$ and a basic open interval $(a-\epsilon, a+\epsilon)$ around $a$. There must be a closed interval $[b, c]$ from the intersection that we used to define $C_q$, such that $[b, c]\subseteq (a-\epsilon, a+\epsilon)$. By choosing a $q'\in (q,1/2)\cap \mathbb{Q}$ close enough to $q$ we may ensure that the "corresponding" $[b', c']$ interval for $C_{q’}$ (if $[b, c]=[p_1(q), p_2(q)]$ for polynomials $p_i$, let $[b', c']=[p_1(q'), p_2(q')]$) is also contained in $(a-\epsilon, a+\epsilon)$. Again, if $q'$ is close enough to $q$ we can ensure that $[b', c']$ does not intersect $C_q\backslash [b, c]$ so that at least one of the endpoints $b'$ or $c'$ does not lie in $C_q$. But clearly this endpoint lies in $C_{q'}$ so that $(a-\epsilon, b+\epsilon)$ contains a point in $X$ which does not lie in $C_q$. Therefore $C_q$ has empty interior in $X$.