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
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 $G$,
a neutral element $e \in G$,
$(-)\cdot (-) \;\colon\; G \times G \to G$,
a function $(-)^{-1} \;\colon\; G \to G$ such that $g \cdot g^{-1} = e = g^{-1} \cdot g$ for all $g \in G$;
a topology $\tau_G \subset P(G)$ giving $G$ the structure of a topological space
such that the operations $(-)^{-1}$ and $(-)\cdot (-)$ 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 $H \subset G$ of a topological group is closed, hence a closed subgroup.
(e.g Arhangel’skii-Tkachenko 08, theorem 1.3.5)
The set of $H$-cosets is a cover of $G$ by disjoint open subsets. One of these cosets is $H$ 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 $C_e \subset G$ of the neutral element. We may assume without restriction of generality that with $g \in C_e$ any element, then also the inverse element $g^{-1} \in C_e$.
For if this is not the case, then we may enlarge $C_e$ by including its inverse elements, and the result is still a compact neighbourhood of the neutral element: Since taking inverse elements $(-)^{-1} \colon G \to G$ 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 $C_e$ is compact, and the union of two compact subspaces is still compact (obviously, otherwise see this prop).
Now for $n \in \mathbb{N}$, write $C_e^n \subset G$ for the image of $\underset{k \in \{1, \cdots n\}}{\prod} C_e \subset \underset{k \in \{1, \cdots, n\}}{\prod} G$ under the iterated group product operation $\underset{k \in \{1, \cdots, n\}}{\prod} G \longrightarrow G$.
Then
is clearly a topological subgroup of $G$.
Observe that each $C_e^n$ is compact. This is because $\underset{k \in \{1, \cdots, n\}}{\prod}C_e$ 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 $H$ is paracompact.
Observe also that the subgroup $H$ is open, because it contains with the interior of $C_e$ a non-empty open subset $Int(C_e) \subset H$ and we may hence write $H$ as a union of open subsets
Finally, as indicated in the proof of Lemma , the cosets of the open subgroup $H$ are all open and partition $G$ 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 $G$ is connected, then there is just one such coset, namely $H$ itself. The argument above thus shows that a connected locally compact topological group is $\sigma$-compact and (by local compactness) also paracompact.
In the general case, all the cosets are homeomorphic to $H$ which we have just shown to be a paracompact group. Thus $G$ is a disjoint union space of paracompact spaces. This is again paracompact (by this prop.).
A topological group $G$ carries two canonical uniformities: a right and left uniformity. The right uniformity consists of entourages $\sim_{l, U}$ where $x \sim_{l, U} y$ if $x y^{-1} \in U$; here $U$ ranges over neighborhoods of the identity that are symmetric: $g \in U \Leftrightarrow g^{-1} \in U$. The left uniformity similarly consists of entourages $\sim_{r, U}$ where $x \sim_{r, U} y$ if $x^{-1} y \in U$. The uniform topology for either coincides with the topology of $G$.
Obviously when $G$ is commutative, the left and right uniformities coincide. They also coincide if $G$ is compact Hausdorff, since in that case there is only one uniformity whose uniform topology reproduces the given topology.
Let $G$, $H$ be topological groups, and equip each with their left uniformities. Let $f: G \to H$ be a group homomorphism.
The following are equivalent:
The map $f$ is continuous at a point of $G$;
The map $f$ is continuous;
The map $f$ is uniformly continuous.
Suppose $f$ is continuous at $g \in G$. Since neighborhoods of a point $x$ are $x$-translates of neighborhoods of the identity $e$, continuity at $g$ means that for all neighborhoods $V$ of $e \in H$, there exists a neighborhood $U$ of $e \in G$ such that
Since $f$ is a homomorphism, it follows immediately from cancellation that $f(U) \subseteq V$. Therefore, for every neighborhood $V$ of $e \in H$, there exists a neighborhood $U$ of $e \in G$ such that
in other words such that $x \sim_U y \Rightarrow f(x) \sim_V f(y)$. Hence $f$ is uniformly continuous with respect to the right uniformity. By similar reasoning, $f$ is uniformly continuous with respect to the right uniformity.
A unitary representation $R$ of a topological group $G$ in a Hilbert space $\mathcal{H}$ is a continuous group homomorphism
where $\mathcal{U}(\mathcal{H})$ is the group of unitary operators on $\mathcal{H}$ with respect to the strong topology.
Here $\mathcal{U}(\mathcal{H})$ 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 $G$, then an unitary representation of $G$ is sometimes called a quantization of $G$. 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 $\mathcal{U}(\mathcal{H})$ is used and not the norm topology, is that only few homomorphisms turn out to be continuous in the norm topology.
Example: let $G$ be a compact Lie group and $L^2(G)$ be the Hilbert space of square integrable measurable functions with respect to its Haar measure. The right regular representation of $G$ on $L^2(G)$ 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,
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 November 16, 2022 at 15:26:53. See the history of this page for a list of all contributions to it.