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:
- Do any such spaces immediately come to mind?
- Are there suggestions for the types of spaces (or perhaps "removal" processes) other than $\mathbb{R}^n$ that might be more accomodating?
- 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.