Several of my students are meeting metric spaces for the first time this term. This being a somewhat lazy Saturday afternoon, I decided to write down my four favourite results about general metric spaces. In no particular order, they are:
- that every metric space is homeomorphic to a bounded space;
- the Banach Contraction Mapping Theorem (and its limerick)
- McShane’s Extension Theorem; and
- some nice facts about the space of probability measures on a metric space.
Read on to find out a little more about these fun little results…
Every Metric Space is Homeomorphic to a Bounded Space
If (ℳ, d) is a metric space, then δ defined by δ(x, y) := d(x, y) ⁄ (1 + d(x, y)) is a metric on ℳ that’s topologically equivalent to d and gives ℳ unit diameter. Hence, every metrizable topological space can be metrized with a bounded metric.
I like this one mostly because of its pedagogical value. I find that when students are first getting to grips with abstract normed, metric and topological spaces, they are prone to making a lot of “category errors” in uttering / writing phrases like
- “So take an open ball about this point [in a topological space]…” (Argh! You need a metric to define balls, and your topological space doesn’t come with a metric… you meant to take an open neighbourhood.)
- “This topological space is bounded.” (Argh! You need a metric before you can say that the maximum distance between any two points in the space is finite.)
- “Take a continuous linear map from this metric space to…” (Argh! Continuity is fine, but to make sense of linearity you need the domain and range to be vector spaces.)
It’s useful to have examples like the above one to remind one that not only is boundedness a metric (not topological) concept, it can’t even be extended in a sensible way to metrizable topological spaces because a space could — and would! — end up being both bounded and unbounded at the same time!
The Banach Contraction Mapping Theorem
I cannot resist stating this one in limerick form:
If ℳ’s a complete metric space
And non-empty, it’s always the case
That if T‘s a contraction,
Then under its action
Exactly one point stays in place.
What’s not to like about a theorem that underwrites the local existence theory of ordinary and partial differential equations, ensures that a map dropped anywhere in the region that it depicts will have exactly one point of the map lying on top of the point that it represents, and comes in a catchy, easy-to-remember rhyme to boot?
McShane’s Extension Theorem
If ℳ is a metric space, E ⊆ ℳ, and T: E → ℝ is continuous with convex modulus of continuity, then T can be extended to all of ℳ without changing the modulus of continuity.
I’ve used this result in my research, but that real reason that I like this one is because it’s so simple to state, yet it’s just not true once you replace ℝ by any old metric space, even if you then try to replace ℳ by something nice like a Banach space. For example, it’s not even true that every continuous function between two copies of the plane ℝ2, where one copy is given the ℓ∞ “max” norm and the other the ℓ1 “Manhattan” norm, can be extended in this way. In general, the theorem does hold if you replace ℝ by any injective metric space — but that’s cheating, because that’s precisely the definition of an injective metric space! The two nice cases that I know of are when
- ℝ is replaced by ℝn with the “max” norm; and
- both spaces are Hilbert spaces, which is the Kirszbraun–Valentine theorem.
The Probability Simplex of a Metric Space
The space ℘(ℳ) of Borel probability measures (with the topology of weak convergence) on a metric space ℳ is separable and metrizable if ℳ is separable, is compact if ℳ is compact, and is Polish if ℳ is Polish.
This one, which I picked up from Parthasarathy’s book on probability in metric spaces, is just spectacularly neat and tidy. It’s very nice that ℘(ℳ) inherits basically all the nice properties of ℳ itself and that one doesn’t have to worry about obscenities like a compact space with a non-compact space of probability measures on it. It’s also worth noting that, when ℳ is separable, the map that assigns to each point a ∈ ℳ the unit point mass (Dirac measure) δa ∈ ℘(ℳ) is an embedding with closed range, and its convex hull is dense in ℘(ℳ), which underwrites the heuristic that ℘(ℳ) is like a generalized triangle or simplex, the corners of which are the unit point masses at each point of ℳ.