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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!)

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!)
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