I am curious about different notions of dimensionality, particularly as it relates to manifolds.
For instance, there is the standard fixed dimensionality of a pure smooth manifold, the Hausdorff dimension, various different notions of Fractal dimension, and other notions of dimension (e.g. , ). Some allow for fractional dimensionality; my question is whether this is possible somehow for manifolds (or rather for some manifold-like objects).
One interesting note in the manifold wiki article mentions the notion of manifolds where the dimensionality changes, by e.g. disjoint unioning of a sphere and line. However, the connected components must have the same dimension, I believe.
But is there a notion of generalized manifold with dimensionality that can be "smoothly varying", in some sense?
My question is partly motivated by the idea that one can easily "imagine" such a construct (e.g. a surface that forms a long cone that thins out into a line). Of course, this would not be a manifold (indeed, in computer science, discrete "manifolds", e.g. meshes or point sets, that do this are called "non-manifold"), but perhaps there is a generalized notion that admits this and analysis?
The most obvious issue is local coordinates are obviously integral in nature. But is there no way to parameterize fractals locally (which are of fractional dimension)?