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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 1 It 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:
- (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 2 The 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 the Gleason lemmas which 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
For every dyadic rational in , we can now define the open sets by setting
for all and .
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 3 A 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 4 Another important example of a local group is that of a group 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 of group 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.