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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 noteshort 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!)

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!)

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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Pete L. Clark
Pete L. Clark
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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\}$$\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!)

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!)

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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Pete L. Clark
Pete L. Clark
  • 65.4k
  • 12
  • 241
  • 381

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!)