| ▲ | arutar 4 days ago | |
Once one exists the realm of differentiable manifolds, it is not really reasonable to talk about a single notion of 'dimension'. Topological dimension is indeed something one can define: e.g. the Koch snowflake [1] or the graph of the Weierstrass function [2] have topological dimension 1. Actually, the first is homeomorphic to the unit circle and the second is homeomorphic to real line. It's great if you are doing topology and you only care about how things look like up to homeomorphism. But if you have metric structure (and you care about it), it is not so useful. Minkowski dimension is certainly easy to define but it has some problems: sets which are "very small" (like a sequence `1/log(n)`) can have Minkowski dimension 1. The article has a minor technical oversight: the limit certainly does not need to exist. Minkowski defined it as the limit supremum of the sequence (actually, he defined it in terms of the decay rate of the size of the neighbourhood of the set, but this is equivalent). But one could analogously define a "lower" variant by taking the limit infimum instead. Hausdorff dimension is not discussed in this article, but it is probably the most "robust" notion of dimension one can define. The Hausdorff dimension of any sequence is 0. But even then, lots of sets with Hausdorff dimension 1 can be very small, like the fat Cantor set which has dimension 1 but has length 0 [3]. So this 'dimension' does not necessarily line up with the intuition for "1-dimensional" in esoteric circumstances. But even Hausdorff / Minkowski dimension does not capture the essence of some matters. For example, one might be interested in when a certain space can be mapped into another space without too much distortion (let's say by a map which respects the metric, like a bi-Lipschitz map). It can easily happen that a set has small (finite) Hausdorff or Minkowski dimension, but it cannot be embedded in a non-distorting way in any finite dimensional Euclidean space. This happens for instance with the real Heisenberg group [4]. If you are interested in this type problem then you want something like Assouad dimension [5]. The moral of the story is: the correct notion of dimension depends critically on what you want to do with your notion of 'dimension'. For sets which are very nice (smooth manifolds) all "reasonable" notions of dimension will coincide with what you expect; but beyond this there is an infinite zoo of ways to define dimension which are all reasonable in various ways, but capture genuinely different notions of 'size'. [1]: https://en.wikipedia.org/wiki/Koch_snowflake [2]: https://en.wikipedia.org/wiki/Weierstrass_function [3]: https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%9... | ||