Universal planar set for countable real sections ($A=\{(c(s),s_n):s\in {\mathbb{R}}^{\mathbb{N}},n\ge 1\}$)

Use coding of real sequences by ternary digits to give each real sequence a distinct column coordinate $c(s)\in [0,1/2]$. Put every term of that sequence in its column. The vertical affine section of a planar set at $c(s)$ is exactly the sequence range. Every nonempty countable set has such an enumeration, with repetitions allowed, while any column outside the code interval gives the empty set. This realizes every countable real subset; the other sections are unrestricted.