A topological space is said to be *metrisable* if one can find a metric on it whose open balls generate the topology.

There are some obvious necessary conditions on the space in order for it to be metrisable. For instance, it must be Hausdorff, since all metric spaces are Hausdorff. It must also be first countable, because every point in a metric space has a countable neighbourhood base of balls , .

In the converse direction, being Hausdorff and first countable is not always enough to guarantee metrisability, for a variety of reasons. For instance the long line is not metrisable despite being both Hausdorff and first countable, due to a failure of paracompactness, which prevents one from gluing together the local metric structures on this line into a global one. Even after adding in paracompactness, this is still not enough; the real line with the lower limit topology (also known as the *Sorgenfrey line*) is Hausdorff, first countable, and paracompact, but still not metrisable (because of a failure of second countability despite being separable).

However, there is one important setting in which the Hausdorff and first countability axioms *do* suffice to give metrisability, and that is the setting of topological groups:

Theorem 1 (Birkhoff-Kakutani theorem)Let be a topological group (i.e. a topological space that is also a group, such that the group operations and are continuous). Then is metrisable if and only if it is both Hausdorff and first countable.

Remark 1It is not hard to show that a topological group is Hausdorff if and only if the singleton set is closed. More generally, in an arbitrary topological group, it is a good exercise to show that the closure of is always a closed normal subgroup of , whose quotient is then a Hausdorff topological group. Because of this, the study of topological groups can usually be reduced immediately to the study of Hausdorff topological groups. (Indeed, in many texts, topological groups are automatically understood to be an abbreviation for “Hausdorff topological group”.)

The standard proof of the Birkhoff-Kakutani theorem (which we have taken from this book of Montgomery and Zippin) relies on the following Urysohn-type lemma:

Lemma 2 (Urysohn-type lemma)Let be a Hausdorff first countable group. Then there exists a bounded continuous function with the following properties:

- (Unique maximum) , and for all .
- (Neighbourhood base) The sets for form a neighbourhood base at the identity.
- (Uniform continuity) For every , there exists an open neighbourhood of the identity such that for all and .

Note that if had a left-invariant metric, then the function would suffice for this lemma, which already gives some indication as to why this lemma is relevant to the Birkhoff-Kakutani theorem.

Let us assume Lemma 2 for now and finish the proof of the Birkhoff-Kakutani theorem. We only prove the difficult direction, namely that a Hausdorff first countable topological group is metrisable. We let be the function from Lemma 2, and define the function by the formula

where is the space of bounded continuous functions on (with the supremum norm) and is the left-translation operator .

Clearly obeys the the identity and symmetry axioms, and the triangle inequality is also immediate. This already makes a pseudometric. In order for to be a genuine metric, what is needed is that have no non-trivial translation invariances, i.e. one has for all . But this follows since attains its maximum at exactly one point, namely the group identity .

To put it another way: because has no non-trivial translation invariances, the left translation action gives an embedding , and then inherits a metric from the metric structure on .

Now we have to check whether the metric actually generates the topology. This amounts to verifying two things. Firstly, that every ball in this metric is open; and secondly, that every open neighbourhood of a point contains a ball .

To verify the former claim, it suffices to show that the map from to is continuous, follows from the uniform continuity hypothesis. The second claim follows easily from the neighbourhood base hypothesis, since if then .

Remark 2The above argument in fact shows that if a group is metrisable, then it admits a left-invariant metric. The idea of using a suitable continuous function to generate a useful metric structure on a topological group is a powerful one, for instance underlying theGleason lemmaswhich are fundamental to the solution of Hilbert’s fifth problem. I hope to return to this topic in a future post.

Now we prove Lemma 2. By first countability, we can find a countable neighbourhood base

of the identity. As is Hausdorff, we must have

Using the continuity of the group axioms, we can recursively find a sequence of nested open neighbourhoods of the identity

such that each is symmetric (i.e. if and only if ), is contained in , and is such that for each . In particular the are also a neighbourhood base of the identity with

For every dyadic rational in , we can now define the open sets by setting

where is the binary expansion of with . By repeated use of the hypothesis we see that the are increasing in ; indeed, we have the inclusion

We now set

with the understanding that if the supremum is over the empty set. One easily verifies using (4) that is continuous, and furthermore obeys the uniform continuity property. The neighbourhood base property follows since the are a neighbourhood base of the identity, and the unique maximum property follows from (3). This proves Lemma 2.

Remark 3A very similar argument to the one above also establishes that every topological group is completely regular.

Notice that the function constructed in the above argument was localised to the set . As such, it is not difficult to localise the Birkhoff-Kakutani theorem to *local groups*. A local group is a topological space equipped with an identity , a *partially defined* inversion operation , and a *partially defined* product operation , where , are open subsets of and , obeying the following restricted versions of the group axioms:

- (Continuity) and are continuous on their domains of definition.
- (Identity) For any , and are well-defined and equal to .
- (Inverse) For any , and are well-defined and equal to . is well-defined and equal to .
- (Local associativity) If are such that , , , and are all well-defined, then .

Informally, one can view a local group as a topological group in which the closure axiom has been almost completely dropped, but with all the other axioms retained. A basic way to generate a local group is to start with an ordinary topological group and restrict it to an open neighbourhood of the identity, with and . However, this is not quite the only way to generate local groups (ultimately because the local associativity axiom does not necessarily imply a (stronger) global associativity axiom in which one considers two different ways to multiply more than three group elements together).

Remark 4Another important example of a local group is that of agroup chunk, in which the sets and are somehow “generic”; for instance, could be an algebraic variety, Zariski-open, and the group operations birational on their domains of definition. This is somewhat analogous to the notion of a “ group” in additive combinatorics. There are a number ofgroup chunk theorems, starting with a theorem of Weil in the algebraic setting, which roughly speaking assert that a generic portion of a group chunk can be identified with the generic portion of a genuine group.

We then have

Theorem 3 (Birkhoff-Kakutani theorem for local groups)Let be a local group which is Hausdorff and first countable. Then there exists an open neighbourhood of the identity which is metrisable.

*Proof:* (Sketch) It is not difficult to see that in a local group , one can find a symmetric neighbourhood of the identity such that the product of any (say) elements of (multiplied together in any order) are well-defined, which effectively allows us to treat elements of as if they belonged to a group for the purposes of simple algebraic manipulation, such as applying the cancellation laws for . Inside this , one can then repeat the previous arguments and eventually end up with a continuous function supported in obeying the conclusions of Lemma 2 (but in the uniform continuity conclusion, one has to restrict to, say, , to avoid issues of ill-definedness). The definition (1) then gives a metric on with the required properties, where we make the convention that vanishes for (say) and .

My motivation for studying local groups is that it turns out that there is a correspondence (first observed by Hrushovski) between the concept of an approximate group in additive combinatorics, and a locally compact local group in topological group theory; I hope to discuss this correspondence further in a subsequent post.

## 11 comments

Comments feed for this article

18 May, 2011 at 5:43 am

oecLemma 2 should probably read: “Let be a Hausdorff first countable _group_”?

[Corrected, thanks – T.]18 May, 2011 at 9:47 am

plmThank you very much for the -extra- efforts in giving references and motivations. It really helps alot -really.

I hope this does not make you feel like a magician giving away his tricks. I think Arne Beurling had a different attitude, and I wonder how mathematicians were/are distributed, with respect to this.

19 May, 2011 at 12:16 am

Marius BuligaLooking forward to learn more about the correspondence between local groups and approximate groups and maybe Gromov’s proof.

Here is why: all information about a (local) group is contained in the group operation (as a function). In metric spaces with dilations (aka dilatation structures or dilation structures), in particular in groups with a left invariant metric endowed with an approximate contraction, the operation of a locally compact local group which is conical (or contractible, like in in particular Carnot groups) appear as a limit of operations which are only approximately associative, so these operations are only approximate group operations.

So, let us compare approximate groups with “approximate groups operations” like those appearing in metric spaces with dilations:

Approximate groups are finite sets in groups which are “approximately closed” with respect to an exact group operation. Under rough equivalence,

i.e. seen from afar in a combinatorial sense, they may be alike local groups which are contractible, then by a theorem of Siebert nilpotent graded groups.

Approximate group operations (induced by the approximate self-similarity of

a metric space with dilations) don’t need an ambient group operation in order to be defined. Instead, in the limit (of a scale parameter), which corresponds to looking at smaller and smaller regions around a point in the metric space, again a la Gromov, these operations converge to a local conical, or contractible group.

How are these related? Probably Hrushovski correspondence between local groups and additive combinatorics cold answer to this, anyway that is what I hope in a part of a joint project with Laurentiu Leustean.

27 May, 2011 at 12:32 pm

Locally compact groups with faithful finite-dimensional representations « What’s new[…] For future reference, we note that this already shows that is first-countable, and hence metrisable, thanks to the Birkhoff-Kakutani theorem as discussed earlier. […]

8 June, 2011 at 5:03 am

Hilbert fifth’s problem without one parameter subgroups | chorasimilarity[…] motivation lies in the project described first time in public here. In fact, one of the reasons to start this blog is to have a place where I can leisurely explain […]

13 June, 2011 at 12:07 pm

Brouwer’s fixed point and invariance of domain theorems, and Hilbert’s fifth problem « What’s new[…] group is of course locally compact Hausdorff; it is also first countable, and hence (by the Birkhoff-Kakutani theorem, discussed in this post) is also metrisable. Because of this, it is not difficult to show that the […]

17 June, 2011 at 2:15 pm

Hilbert’s fifth problem and Gleason metrics « What’s new[…] bump functions with which to create a metric, in a variant of the analysis used to prove the Birkhoff-Kakutani theorem); Theorem 5 also requires Peter-Weyl theorem (to dispose of certain compact subgroups that arise en […]

12 July, 2011 at 5:53 pm

Gleason’s lemma « What’s new[…] construction was already seen in the proof of the Birkhoff-Kakutani theorem, which is the main tool used to establish Proposition 6. For the other propositions, the idea is to […]

17 August, 2011 at 7:04 pm

Notes on local groups « What’s new[…] a global topological group is first-countable if and only if it is locally first countable. (By the Birkhoff-Kakutani theorem, this implies that such groups are metrisable if and only if they are locally […]

2 May, 2012 at 2:14 pm

Kasper HenriksenI take it that the theorem is stated as is because the slightly more general “a topological group is pseudometrizable if and only if it is first countable” (from which the Birkhoff-Kakutani theorem would follow by quotienting out the closure of the identity) is false

3 May, 2012 at 1:19 pm

Terence TaoActually, that latter formulation is equivalent to the one stated here. Given that the reduction to the Hausdorff case is not too difficult, and that metrics are more familiar than pseudometrics, I did not state the theorem in the maximally general formulation.