topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
A topological group is a topological space with a continuous group structure: a group object internal to the category Top of topological spaces and continuous functions between them.
A topological group is
a group, hence
a set ,
a neutral element ,
a function such that for all ;
a topology giving the structure of a topological space
such that the operations and are continuous functions (the latter with respect to the product topology).
Sometimes topological groups are understood to be Hausdorff (e.g. Bredon 72, p. 1).
(open subgroups of topological groups are closed)
Every open subgroup of a topological group is closed, hence a closed subgroup.
(e.g Arhangel’skii-Tkachenko 08, theorem 1.3.5)
The set of -cosets is a cover of by disjoint open subsets. One of these cosets is itself and hence it is the complement of the union of the other cosets, hence the complement of an open subspace, hence closed.
(connected locally compact topological groups are sigma-compact)
Every connected locally compact topological group is sigma-compact.
Every locally compact topological group is paracompact.
(e.g. Arhangel’skii-Tkachenko 08, cor. 3.1.4, cor. 3.1.5)
By assumption of local compactness, there exists a compact neighbourhood of the neutral element. We may assume without restriction of generality that with any element, then also the inverse element .
For if this is not the case, then we may enlarge by including its inverse elements, and the result is still a compact neighbourhood of the neutral element: Since taking inverse elements is a continuous function, and since continuous images of compact spaces are compact, it follows that also the set of inverse elements to elements in is compact, and the union of two compact subspaces is still compact (obviously, otherwise see this prop).
Now for , write for the image of under the iterated group product operation .
Then
is clearly a topological subgroup of .
Observe that each is compact. This is because is compact by the Tychonoff theorem, and since continuous images of compact spaces are compact. Thus
is a countable union of compact subspaces, making it sigma-compact. Since locally compact and sigma-compact spaces are paracompact, this implies that is paracompact.
Observe also that the subgroup is open, because it contains with the interior of a non-empty open subset and we may hence write as a union of open subsets
Finally, as indicated in the proof of Lemma , the cosets of the open subgroup are all open and partition as a disjoint union space (coproduct in Top) of these open cosets. From this we may draw the following conclusions.
In the particular case where is connected, then there is just one such coset, namely itself. The argument above thus shows that a connected locally compact topological group is -compact and (by local compactness) also paracompact.
In the general case, all the cosets are homeomorphic to which we have just shown to be a paracompact group. Thus is a disjoint union space of paracompact spaces. This is again paracompact (by this prop.).
A topological group carries two canonical uniformities: a right and left uniformity. The right uniformity consists of entourages where if ; here ranges over neighborhoods of the identity that are symmetric: . The left uniformity similarly consists of entourages where if . The uniform topology for either coincides with the topology of .
Obviously when is commutative, the left and right uniformities coincide. They also coincide if is compact Hausdorff, since in that case there is only one uniformity whose uniform topology reproduces the given topology.
Let , be topological groups, and equip each with their left uniformities. Let be a group homomorphism.
The following are equivalent:
The map is continuous at a point of ;
The map is continuous;
The map is uniformly continuous.
Suppose is continuous at . Since neighborhoods of a point are -translates of neighborhoods of the identity , continuity at means that for all neighborhoods of , there exists a neighborhood of such that
Since is a homomorphism, it follows immediately from cancellation that . Therefore, for every neighborhood of , there exists a neighborhood of such that
in other words such that . Hence is uniformly continuous with respect to the right uniformity. By similar reasoning, is uniformly continuous with respect to the right uniformity.
A unitary representation of a topological group in a Hilbert space is a continuous group homomorphism
where is the group of unitary operators on with respect to the strong topology.
Here is a complete, metrizable topological group in the strong topology, see (Schottenloher, prop. 3.11).
In physics, when a classical mechanical system is symmetric, i.e. invariant in a proper sense, with respect to the action of a topological group , then an unitary representation of is sometimes called a quantization of . See at geometric quantization and orbit method for more on this.
The reason that in the definition of a unitary representation, the strong operator topology on is used and not the norm topology, is that only few homomorphisms turn out to be continuous in the norm topology.
Example: let be a compact Lie group and be the Hilbert space of square integrable measurable functions with respect to its Haar measure. The right regular representation of on is defined as
and this will generally not be continuous in the norm topology, but is always continuous in the strong topology.
Hilbert's fifth problem: “Which topological groups admit Lie group structure?”
The category TopGrp of topological groups and continuous group homomorphisms between them is a protomodular category.
A proof is spelled out by Todd Trimble here on MO.
The classical Lie groups are in particular topological groups, such as the general linear group and its subgroups.
…
topological group,
Nicolas Bourbaki, Topological Groups, Chapter 3 in: General topology, Elements of Mathematics III, Springer (1971, 1990, 1995) [doi:10.1007/978-3-642-61701-0]
Glen Bredon, Chapter 0 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)
Alexander Arhangel’skii, Mikhail Tkachenko, Topological Groups and Related Structures, Atlantis Press 2008 (doi:10.2991/978-94-91216-35-0)
The following monograph is not particulary about group representations, but some content of this page is based on it:
Last revised on March 28, 2024 at 13:35:38. See the history of this page for a list of all contributions to it.