What properties of metric spaces are not preserved by uniformly continuous isomorphism?

Question

Compactness and connectedness are preserved by homeomorphism, in the sense that if two metric spaces $(X,d_X)$ and $(Y,d_Y)$ are homeomorphic and $(X,d_X)$ is compact then it follows that $(Y,d_Y)$ is also compact (and similarly for connectedness). However, if $(X,d_X)$ is complete it does not necessarily follow that $(X,d_Y)$ is also complete (to see this, we can take $X = \mathbb{N}$ and $Y = \{\frac{1}{n} : n \in \mathbb{N}\setminus\{0\}\}$, both with the usual metric). In fact the same counterexample can be used to show that boundedness is not preserved by homeomorphism.

On the other hand, it can be shown that uniformly continuous isomorphism preserves compactness, connectedness, completeness and boundedness, in the sense that if there exists a uniformly continuous bijection $f:(X,d_X) \rightarrow (Y,d_Y)$ whose inverse is uniformly continuous and $(X,d_X)$ has one of the properties listed earlier, than $(Y,d_Y)$ also has that property.

Compactness, connectedness, completeness and boundedness are all properties which seem "important" in some sense in the context of metric spaces. So are there any other "important" properties that metric spaces can have which are not preserved by uniformly continuous isomorphism?

Answer 1

You probably do not know any of the properties/invariants listed below, but these are important in the theory of metric spaces. Here are some examples:

1. Hausdorff dimension is not preserved by uniformly continuous isomorphisms. For instance, you can have a compact metric space $(X_1,d_1)$ of zero Hausdorff dimension and a compact metric space $(X_2,d_2)$ of infinite Hausdorff dimension, such that $X_1$ is homeomorphic to $X_2$.

2. A metric space $(X,d)$ is called a path-metric space if for any two points $x,y\in X$, $d(x,y)$ equals infimum of lengths of paths in $X$ connecting $x$ to $y$. (Another name is an intrinsic metric space.) You can have two homeomorphic metric compacts $(X_1,d_1)$, $(X_2,d_2)$ such that the first one is a path-metric space while the second contains no rectifiable nonconstant paths.

3. You can have two countably-infinite metric spaces $(X_1,d_1)$, $(X_2,d_2)$ both of which have discrete topology and satisfy $d_i(x,y)\ge 1$ for all $x\ne y$, such that the first one is hyperbolic in the sense of Gromov (where hyperbolicity is defined via Gromov-products) and the second is not. But, $(X_1,d_1)$, $(X_2,d_2)$ are isomorphic via a uniformly continuous homeomorphism (any bijection $X_1\to X_2$ will do the job).

4. Quasi-isometry type is not preserved by a uniformly continuous homeomorphism: You can have two countably-infinite metric spaces $(X_1,d_1)$, $(X_2,d_2)$ both of which have discrete topology and satisfy $d_i(x,y)\ge 1$ for all $x\ne y$, such that $(X_1,d_1)$, $(X_2,d_2)$ are not quasi-isometric to each other. However, as in (3), there is a uniformly continuous isomorphism of these metric spaces.