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I recently completed a short expository note on this subject, Covering Numbers in Linear Algebra. See:

http://math.uga.edu/~pete/coveringnumbersv2.pdf

I recently completed a short expository note on this subject, Covering Numbers in Linear Algebra. See:

http://alpha.math.uga.edu/~pete/coveringnumbersv2.pdf

Sometime around the summer of 2008 I had a week in which I was downright obsessed with this question and its various analogues and generalizations. Here is the written record of that obsession:

http://www.math.uga.edu/~pete/closedsubsets.pdf

Warning: the title makes a promise that the paper does not keep -- I never got to writing up arithmetic geometric versions of this problem. Overall, I can detect the mania in the fact that the manuscript is globally quite disorganized but a lot of the exposition (mostly of easy things) is highly polished and even somewhat flowery.

Nevertheless, I think there's some interesting [not fascinating] stuff in there, including possibly some avenues of [not earth-shattering] future research. Please feel free to contact me if you have any suggestions.

Addendum: OK, just to sell it a bit, here is the relevant result that I prove:

Let $V$ be a vector space over a field $K$, of dimension at least $2$. A linear covering of $V$ is a family of proper linear subspaces of $V$ whose union is $V$. The linear covering number of $V$ is the least cardinality of a linear covering.

Theorem: a) If at least one of $\# K$, $\operatorname{dim}(V)$ is finite, the linear covering number of $V$ is $\# K + 1$.
b) If both $\# K$ and $\operatorname{dim}(V)$ are infinite, the linear covering number of $V$ is $\aleph_0$.

There is also the following amusing fact: say a covering is irredundant if no proper subset gives a covering. It is very tempting to think that the linear covering number can be attained by an irredundant covering. But in fact the minimal cardinality of an irredundant covering is always $\# K + 1$, so e.g. redundant coverings of infinite dimensional $\mathbb{R}$-vector spaces are much more efficient than irredundant coverings! (This is not so hard to prove, once you realize it's true.)

In the (incomplete) remainder of the manuscript, I consider various analogues of the linear covering number: vector spaces by affine linear subspaces, groups by subgroups, groups by cosets, topological spaces by closed subspaces, algebraic varieties by subvarieties...

I recently completed a short expository note on this subject, Covering Numbers in Linear Algebra. See:

http://math.uga.edu/~pete/coveringnumbersv2.pdf

Sometime around the summer of 2008 I had a week in which I was downright obsessed with this question and its various analogues and generalizations. Here is the written record of that obsession:

http://www.math.uga.edu/~pete/closedsubsets.pdf

Warning: the title makes a promise that the paper does not keep -- I never got to writing up arithmetic geometric versions of this problem. Overall, I can detect the mania in the fact that the manuscript is globally quite disorganized but a lot of the exposition (mostly of easy things) is highly polished and even somewhat flowery.

Nevertheless, I think there's some interesting [not fascinating] stuff in there, including possibly some avenues of [not earth-shattering] future research. Please feel free to contact me if you have any suggestions.

Sometime around the summer of 2008 I had a week in which I was downright obsessed with this question and its various analogues and generalizations. Here is the written record of that obsession:

http://www.math.uga.edu/~pete/closedsubsets.pdf

Warning: the title makes a promise that the paper does not keep -- I never got to writing up arithmetic geometric versions of this problem. Overall, I can detect the mania in the fact that the manuscript is globally quite disorganized but a lot of the exposition (mostly of easy things) is highly polished and even somewhat flowery.

Nevertheless, I think there's some interesting [not fascinating] stuff in there, including possibly some avenues of [not earth-shattering] future research. Please feel free to contact me if you have any suggestions.

Addendum: OK, just to sell it a bit, here is the relevant result that I prove:

Let $V$ be a vector space over a field $K$, of dimension at least $2$. A linear covering of $V$ is a family of proper linear subspaces of $V$ whose union is $V$. The linear covering number of $V$ is the least cardinality of a linear covering.

Theorem: a) If at least one of $\# K$, $\operatorname{dim}(V)$ is finite, the linear covering number of $V$ is $\# K + 1$.
b) If both $\# K$ and $\operatorname{dim}(V)$ are infinite, the linear covering number of $V$ is $\aleph_0$.

There is also the following amusing fact: say a covering is irredundant if no proper subset gives a covering. It is very tempting to think that the linear covering number can be attained by an irredundant covering. But in fact the minimal cardinality of an irredundant covering is always $\# K + 1$, so e.g. redundant coverings of infinite dimensional $\mathbb{R}$-vector spaces are much more efficient than irredundant coverings! (This is not so hard to prove, once you realize it's true.)

In the (incomplete) remainder of the manuscript, I consider various analogues of the linear covering number: vector spaces by affine linear subspaces, groups by subgroups, groups by cosets, topological spaces by closed subspaces, algebraic varieties by subvarieties...