We’re going to talk about an interesting merger of abstract algebra and topology, namely topological groups.
Topological algebraic structures emerg all the time in mathematics. For example, in calculus you were actually working with the topological field . The topology of
as well as its algebraic properties were used to prove statements about differentiation and integration. There’s no better place to start than with definitions, so let’s define a topological group and introduce some of its basic properties.
The Definition of A Topological Group and Some Consequences
A topological group is a group
endowed with a topology such that the group operation and inverse map
are continuous. Here we give the product topology. By convention, if
is finite it’s given the discrete topology.
A nice property of groups is that we can move between any two elements in the group by translation (i.e. multiplication). We would hope that if is open, then
and
are open for any
. This is indeed true, and to see it start by fixing
. Then the map
is obviously continuous. Composing with the group operation, the map
is continuous. This composition is said to be a left-translation or left-translation by
. Since
is a continuous inverse, left-translation is a homeomorphism. There is a symmetric argument for right-translation. So, the above tells us in particular that if
is open, then so is
and
. We’ve neglected the inverse map
up until now, but it has an analogous property as the above translation maps do. It’s a continuous involution, so is a homeomorphism. Therefore if
is open, then the image of
under the inverse map is open, and we call it the inverse of
and denote it by
.
Another topological property which is important for topological groups is homogeneity. If is a topological space, then
is homogeneous if for any
there is a homeomorphism
such that
. Intuitively, this says that the topology of
looks the same locally any any point
. If we look at a topological group
, then it’s homogeneous because if
, then the left-translation map
sends
to
. A consequence of this is that if we have a local basis at the identity
, then it determines a local basis everywhere and therefore a basis for the entire topology. Moreover, homogeneity says that it suffices to verify local topological properties of
only at the identity
. Indeed, if we have such a property at
we can then translate this property everywhere in
. We will see an example of this below.
Symmetric Neighborhoods and The Equivalence of
and 
An interesting property of topological groups is that the and
(Hausdorff) separation axioms are equivalent! In the following we will prove this statement, but we will need a theorem first which is incredibly useful beyond its introduction here. This theorem is about symmetric neighborhoods. Firstly, a set
is called symmetric if
. In other words, the set is closed under inverses. Now for the theorem:
Theorem: Letbe a topological group. Every neighborhood
of the identity contains a symmetric neighborhood
of the identity such that
.
Proof. We may assume is open because if not, we take its interior. The restriction of the group operation
is continuous, so
is an open set containing
of the form
for some open neighborhoods
and
of
. If we set
, then
is a neighborhood of
such that
. Now observe that
and
are both neighborhoods of the identity. Set
. Then
is the desired neighborhood since
and
is symmetric by construction.
By our homogeneity discussion above, the proposition holds for any , not just the identity. Now that we have the technical proposition, let’s state and prove the equivalence of
and
.
Theorem: Letbe a topological group. Then
is
if and only if it's
.
Proof. The forward implication is clear since every space is
. To prove the inverse implication, let
. Since
is
we can find a neighborhood
about the identity disjoint from
, and we may assume
is open by replacing it with its interior. By the proposition, there is a symmetric neighborhood
, about the identity, contained in
, and such that
. Again, we may assume
is open by replacing it with its interior. We claim the open sets
and
are disjoint. If they are not, we have
for some
which implies
, a contradiction. Hence
is
.
It’s a very easy exercise to show these equivalent statements are further equivalent to the conditions of the identity being closed and every singleton of being closed. It’s also a fun exercise to show that the general linear group over the reals
is a topological group with respect to matrix multiplication and the Euclidean topology (here you view matrices as
dimensional vectors in
), and that its special linear group
is a closed subgroup.
References
Fourier Analysis on Number Fields – Dinakar Ramakrishnan, Robert J. Valenza