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If you consider $\mathbb{R}^3$ and a one-dimensional space curve, by "removing" the curve from $\mathbb{R}^3$ you are left with a space that is still three dimensional, for an appropriate definition of dimension (perhaps the one from linear algebra?). Thus the naive equation 3D - 1D = 2D does not hold.

I am interested in spaces or sets where such an equation would hold. That is, by removing a set of a given dimension, the total space must be arbitrarily reduced by that dimension.

My question has three parts:

  1. Do any such spaces immediately come to mind?
  2. Are there suggestions for the types of spaces (or perhaps "removal" processes) other than $\mathbb{R}^n$ that might be more accomodating?
  3. Am I using too naive a definition of dimension? I am most familiar with dimension as the cardinality of the set of basis vectors in a vector space, but am aware of a Hausdorff dimension for topological spaces, if that's where I should turn...

My apologies if this question makes no sense, or is too poorly thought out to receive adequate response.

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    Yes, I had thought to quotient instead (and probably should have mentioned it above). I guess that removes it from the purview of "arithmetic"? Qiaochu touches on this in his great answer below...2011-07-08

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This should never be true for a reasonable definition of dimension (for example the dimension of a manifold). A lower-dimensional thing should have measure zero in a higher-dimensional thing, so removing it shouldn't change the dimension of the higher-dimensional thing.

The correct version of the "naive equation" is that the Cartesian product of an $m$-dimensional thing and an $n$-dimensional thing should be an $m+n$-dimensional thing. Putting two things together (the opposite of removing a thing from another thing) doesn't add their dimensions; instead, the disjoint union of an $m$-dimensional thing and an $n$-dimensional thing should be considered to have dimension $\text{max}(m, n)$.

This algebraic structure (with addition and max instead of multiplication and addition) happens to have a name: it's called the max-plus semiring.

(If you really want to think about subtraction of dimensions instead of addition, then I guess the appropriate thing is to take the quotient by the free action of a Lie group. In nice cases, it should be true that the quotient of an $m$-dimensional manifold by an $n$-dimensional Lie group acting freely has dimension $m-n$. The easiest case of this occurs when we take the quotient of a vector space by a subspace.)

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    @Jason: ah, you're right, I want a free action.2011-07-08