Let $V$ be a vector space over a field $F$, of dimension at least $2$, and consider coverings $\mathcal{C} = \{W_i\}$(W_i)$of$V$by proper subspaces. Does there exist a countable covering$\mathcal{C}$? (It depends on$\dim V$and $\# F$, but perhaps not exactly as you expect!) 1 [made Community Wiki] This is of the "to puzzle graduate students" variety, but I was taken enough with it to write it up in a short note: Let$V$be a vector space over a field$F$, of dimension at least$2$, and consider coverings $\mathcal{C} = \{W_i\}$ of$V$by proper subspaces. Does there exist a countable covering$\mathcal{C}$? (It depends on$\dim V$and $\# F\$, but perhaps not exactly as you expect!)