 Baire avoidance of countably many affine hyperplanes

Every proper affine hyperplane in ${\mathbb{R}}^n$ is closed and has empty interior: a small displacement along its nonzero normal vector leaves it. The Baire category theorem therefore makes the complement of a countable collection of such hyperplanes dense. In particular, avoiding the hyperplanes ${\sum }_{j=1}^n{k}_j{x}_j=k_{n+1}$ for nonzero integer tuples yields a vector for which $1,x_1,\dots ,x_n$ are linearly independent over $\mathbb{Q}$. Tuples with all the first $n$ entries zero define an empty forbidden set.