show/hide this revision's text 2 added 2 characters in body; deleted 6 characters in body

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 $\mathcal{C} = \{W_i\}$(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!)
show/hide this revision's text 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!)