Simple examples of proper metric spaces?

Question

I've encountered the term of a "proper" metric space(a metric space is called proper if every closed, bounded subspace is compact), which struck as quite an interesting one, but I can't find any good examples other than $ \mathbb{R}^n $. I've come across this paper: http://www.math.ku.dk/~haagerup/publications/proper_metric_preamble.pdf but it seems to require a decent of knowledge of alebraic topology, which I have no clue about.

Are there any fairly elementary examples of such spaces?

Answer 1

So, examples should be simple and good. Two undefined requirements... Take your pick.

1. Every finite set, with an arbitrary metric.
2. Every closed polygon in the plane: triangle, square.
3. Every closed polyhedron in $\mathbb R^3$.
4. More generally, every closed subset of $\mathbb R^n$.
5. Yet more generally, the image of a closed subset of $\mathbb R^n$ under a bi-Lipschitz map.
6. Every graph (possibly infinite) of bounded degree, with path metric (distance between two vertices is the length of shortest path between them). In particular, the Cayley graph of a finitely generated group.
7. Every complete Riemannian manifold: sphere, torus, Klein bottle, projective space...
8. Every complete Finsler manifold.
9. Every complete sub-Riemannian or sub-Finsler manifold.
10. A finite product of any of the above.

Loosely speaking, if you come across a metric space that is

• complete, and
• does not come from functional analysis

then it's probably proper.