If F is a Continuous Bijection Then F is a Homemomorphism

Continuous Bijection

Any continuous bijection between completely regular spaces with Baire measurableinverse transforms a lifting compact space into a lifting compact one.

From: Handbook of Measure Theory , 2002

Liftings

Werner Strauss , ... Kazimierz Musiat , in Handbook of Measure Theory, 2002

THEOREM 7.1

(i)

An arbitrary subspace of a compact metrizable space is lifting compact.

(ii)

Every Baire set in a lifting compact space is lifting compact.

(iii)

Any continuous bijection between completely regular spaces with Baire measurableinverse transforms a lifting compact space into a lifting compact one.

(iv)

A countable product of lifting compact spaces is lifting compact.

The results (ii) to (iv) are analogs of results proved by Moran (1969) for (strongly) measure compact spaces. A Banach space E under its weak topology is lifting compact if and only if every E-valued scalarly measurable function is scalarly equivalent to a Bochner measurable function by Bellow (1980, section 6, Remark 2). Every subspace of a compact metric space is strongly lifting compact. Moreover, this holds true for every strongly lifting compact space. The next three theorems are taken from Babiker et al. (1986).

THEOREM 7.2

(i)

An arbitrary subset of a strongly lifting compact space is strongly lifting compact.

(ii)

Countable unions of strongly lifting compact Baire subsets of a completely regular Hausdorff space are strongly lifting compact.

(iii)

A countable product of strongly lifting compact spaces is strongly lifting compact.

(iv)

The image of a strongly lifting compact space under a continuous surjection with Baire measurable section is strongly lifting compact.

(v)

A measure compact space is strongly lifting compact if every point has a strongly measure compact neighborhood.

The next result states a relation between strong lifting compactness and the USLP. Its proof relies on Theorem 3.7.

THEOREM 7.3

If the map f: Ω → T is strongly lifting compact and υ is the image measure of μ under f on the α-algebra $B:=\left\{B\subseteq T:f^{-1}\left(B\right)\in \Sigma \right\},$ then the topological measure space (T, T, B, υ) has the USLP, if T denotes the completely regular Hausdorff topology of T.

From the last theorem together with the existence of a compact Radon probability space without strong lifting (see section 4) follows that neither lifting compactness nor strong measure compactness imply strong lifting compactness, nor does strong lifting compactness imply strong measure compactness, as witnessed by the standard Lebesgue non-measurable subset of [0, 1]. For general metric spaces strong lifting compactness is equivalent with measure compactness, which means in that case that every closed discrete subspace has non-measurable cardinal.

It is an open problem, whether the converse of Theorem 7.3 is true, compare Macheras and Strauss (1992). The next result gives among other equivalent conditions an answer to the positive for (E, weak), a metrizable locally convex spaces E under its weak topology. An essential tool for its proof is A. Tortrat's Theorem 8, from Tortrat (1975).

THEOREM 7.4

For a metrizable locally convex space E the following conditions are all equivalent.

(i)

(E, weak) is strongly lifting compact.

(ii)

Every Baire probability measure space based on (E, weak) has the USLP.

(iii)

Every Baire probability measure space based on (E, weak) has the ASLP.

(iv)

(E, weak) is completion regular and measure compact.

(v)

Every Baire probability measure μ on (E, weak) is supported by aμ-measurable closed linear subspace of E which is separable with respect to the metric of E.

(vi)

(E, weak) is measure compact and every Borei subset of (E, metric) is measurable with respect to any Baire probability measure on (E, weak).

(vii)

Every scalarly measurable function from a complete probability space into E agrees a.e. with a Bochner measurable function.

(viii)

(E, weak) is measure compact and {0} is a Baire subset of E with respect to (E, weak).*

(ix)

(E, weak) is measure compact and there exists a sequence in E', the topological conjugate of E, which separates the points of E.

(x)

(E,weak) is measure compact and there is a continuous linear injection from (E, metric) into ℝ^ℕ

(xi)

(E, weak) is measure compact and submetrizable. If the locally convex space E is normable, we may add the following condition.

(xii)

(E, weak) is measure compact and there is a continuous linear injection from (E, norm) into l^∞(N)

It can be seen from the last theorem that within the class of all metrizable locally convex spaces strongly lifting compact spaces can be characterized in a purely topological way. In Strauss (1992) the strong lifting compactness of conjugate Banach spaces under their weak^* topology has been discussed. In this class the equivalence of the condition (vii) in the last theorem with strong lifting compactness as well as with the conditions corresponding to conditions (ii) respectively (iii) of the last theorem breaks down. There are similar characterizations for strongly lifting compact functions in Babiker et al. (1986). The conjugate M([0, 1]) of C_b,([0, 1]) is a non-separable Banach space which is submetrizable under its weak topology. The mild set-theoretic assumption that the continuum is measure compact implies that (M([0, 1]), weak) is measure compact. Therefore by the last theorem M([0, 1]) is an example of a non-separable strongly lifting compact space.

In connection with the definition at the beginning of this section one should mention the following construction of a lifting of Banach space valued functions, which is commonly used in the context of differentiation of vector measures and integration of vector functions. If X is a Banach space and X' is the space of continuous functionals on X then for given function f : Ω → X' satisfying $〈x,f〉\in L^{\infty }\left(\mu \right),$ where by definition $〈x,f〉\left(\omega \right):=\left(f\left(\omega \right)\right)\left(x\right)$ for every ω ∈ Ω and for every x ∈ X, we can for each $\rho \in \Lambda \left(\mu \right)$ define a function $\rho \left(f\right):\Omega \to X^{\prime }$ by setting $〈x,\rho \left(f\right)\left(\omega \right)〉:=\rho \left(〈x,f〉\right)\left(\omega \right)$ where x ∈ X for all ω X for all ω Ω. Von Weizsäcker (1978) proved that ρ(f) is measurable when X' is equipped with its weak^* -topology. This function is an essential tool in investigating several aspects of integration. More details and abandon references can be found in Musial (2002).

In fact we have the following result equivalent to the existence of a lifting (compare A. and C. Ionescu Tulcea (1962, 1969a) and Kölzow (1968)):

THEOREM 7.5

For given c.l.d. measure spaces (Ω, ∑, μ the following conditions are all equivalent:

(i)

There exists a lifting on ${ℒ}^{\infty }\left(\mu \right):$

(ii)

For any normed space X and any bounded linear operator u from L¹ (μ) into X' (the topological dual space of X) there exists a map f from Ω into X' with $〈x,f〉\in {ℒ}^{\infty }\left(\mu \right)$ for all x ∈ X and $\Vert f\Vert ⩽\Vert u\Vert$ such that ${\displaystyle \int 〈x,f〉gd\mu =〈x,u\left(g^{•}\right)〉}$ for all x ∈ X and all $g\in {ℒ}^1\left(\mu \right).$

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Differential and Analytic Manifolds

Henri Bourlès , in Fundamentals of Advanced Mathematics V3, 2019

2.3.3 Submanifolds

Let M be a manifold and N a non-empty subset of M. Suppose that, at every point y ∈ N, there exists a chart (V, ψ, E ₁ × E ₂) of M centered on y such that ψ (V) = W ₁ × W ₂, where W _i is an open set of the Banach space E _i (i = 1,2) and

$\psi \left(N\cap V\right)=W_1\times \left\{0\right\}.$

Then:

–

N is locally closed in M, i.e. every y ∈ N has an open neighborhood V in M such that V  ∩ N is closed in V. Furthermore, ψ induces a continuous bijection ψ ₁  : N  ∩ V  → W ₁.

–

The thus obtained collection of triples (N  ∩ V, ψ ₁, E ₁) forms an atlas of N of class C ^r .

Definition 2.41

When equipped with this atlas, the set N is called a submanifold of M.

In particular, every non-empty open subset of M is a submanifold of M. Any curve in M is a one-dimensional submanifold of M.

Corollary-Definition 2.42

Let M be a manifold, N a submanifold of M and ι : N → M the canonical injection. With the above notation, for all y ∈ N, we have:

$T_y\left(M\right)={\mathbf{E}}_1\times {\mathbf{E}}_2,T_y\left(N\right)={\mathbf{E}}_1,$

so the mapping T _y (ι): T _y (N) → T _y (M) is injective and T _y (M) /T _y (N) ≅= E ₂ is a Banach space, said to be transversal to N (in M) at y. The dimension n ₂ of this space (finite or infinite) is said to be the codimension of N in M at the point y.

Corollary-Definition 2.42 further implies that:

$0\to T_y\left(N\right)\stackrel{T_y\left(\iota \right)}{\to }{T}_y\left(M\right)\stackrel{\phi }{\to }{T}_y\left(M\right)/T_y\left(N\right)\to 0$

is a short exact sequence that splits in Vec ([P1], section 3.1.4(I), Lemma-Definition 3.15), where φ denotes the canonical surjection.

Consider the special case where M and N are fini(e-dimensional. Each point y of N has a neighborhood V in M with local coordinates (ξ¹,…, ξ ^m ) such that the m – n last coordinates of the points of N in V are zero. These local coordinates are therefore of the form:

$\left({\xi }^1,\dots ,{\xi }^n,0,\dots ,0\right)$

and the codimension of N in M is m – n = dim _y (M) – dim _y (N).

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Decomposition Theory

In C*-Algebras and their Automorphism Groups (Second Edition), 2018

4.6.10 Lemma

Let T be a Polish space. For each Borel subset E of T, there is a Polish space $T_1$ and a continuous bijection $f:T_1\to E$ .

Proof

Let $\mathcal{P}$ be the class of Borel sets E in T that satisfy the condition in the lemma. By 4.2.2 $\mathcal{P}$ contains all open sets and all closed sets. Let $\{E_n\}$ be a sequence in $\mathcal{P}$ consisting of pairwise disjoint sets. It is clear that $\bigcup E_n\in \mathcal{P}$ . Let $\{E_n\}$ be an arbitrary sequence from $\mathcal{P}$ and choose Polish spaces $T_n$ and continuous bijections $f_n:T_n\to E_n$ . The subspace F of the Polish space $\prod T_n$ consisting of the points $(t_n)$ such that $f_n(t_n)=f_1(t_1)$ for all n is closed. Projection onto the first coordinate followed by $f_1$ is a continuous bijection of F onto $\bigcap E_n$ . It follows that $\bigcap E_n\in \mathcal{P}$ . Thus $\mathcal{P}$ is closed under intersections and disjoint unions.

Let ${\mathcal{P}}_0$ denote the subset of $\mathcal{P}$ consisting of those sets E in $\mathcal{P}$ for which also $T\backslash E\in \mathcal{P}$ . We see that ${\mathcal{P}}_0$ contains the open sets and that it is closed under complementation. The proof is completed when we show that ${\mathcal{P}}_0$ contains all Borel sets, and for this, we only need to verify that ${\mathcal{P}}_0$ is closed under countable intersections. So take $\{E_n\}$ in ${\mathcal{P}}_0$ . We proved above that $\bigcap E_n\in \mathcal{P}$ . However,

$T\backslash \bigcap E_n=\bigcup T\backslash E_n=\bigcup _n(T\backslash E_n)\cap \left(\bigcap _{k=1}^{n-1}{E}_k\right),$

which is a disjoint union of sets from $\mathcal{P}$ , so that $T\backslash \bigcap E_n\in \mathcal{P}$ , and consequently $\bigcap E_n\in {\mathcal{P}}_0$ . □

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Infinite Words

In Pure and Applied Mathematics, 2004

Proof

Let $\mathcal{U}$ be an A-universal class for Γ(E). Then $\mathcal{U}$ ^c is an A-universal class for ¬Γ(E), since

$\neg \Gamma \left(E\right)=\left\{{\left(\mathcal{U}\left(y\right)\right)}^c|y\in A^{\omega }\right\}=\left\{{\mathcal{U}}^c\left(y\right)|y\in A^{\omega }\right\}$

Let (n, m) → 〈n, m〉 be a fixed bijection from ℕ × ℕ onto ℕ. For each y ∈ A^ω , define $\overline{y}$ _n ∈ A^ω by $\overline{y}$ _n (m) = y(〈n, m〉). We claim that the map y → ( $\overline{y}$ _n ) _n∈ℕ defines a uniformly continuous bijection from A^ω onto (A^ω ) ^ω . First, the map is bijective, since for any sequence (y_n ) _n∈ℕ of elements of A^ω , there is y ∈ A^ω such that $\overline{y}$ _n = y_n for all n ∈ ℕ. Furthermore, given k > 0, let

$K=\max \left\{〈i,j〉|i\le k,j\le k\right\}$

Let now x and y be two elements of A ^ω . If d(x, y) ≤ 2 ^−K+1, that is, if x(0) = y(0), …, x(K) = y(K), then x(〈i, j〉) = y(〈i, j〉) for 0 ≤ i, j ≤ k. It follows that d( $\overline{x}$ ₀, $\overline{y}$ ₀) ≤ 2^−(k+1), …, d( $\overline{x}$ _k , $\overline{y}$ _k ) ≤ 2 ^−(k+1), whence

$d({({\overline{x}}_n)}_{n\in \mathbb{N},}{({\overline{y}}_n)}_{n\in \mathbb{N}})=\sum _{n=0}^{\infty }{2}^{-n}{d}_n({\overline{x}}_n,{\overline{y}}_n)\le 2^{-k}$

which proves uniform continuity.

We now show that the set

${\exists }^{\omega }\mathcal{U}=\left\{\left(y,x\right)\in A^{\omega }\times E|\exists n\in \mathbb{N}\left({\overline{y}}_n,x\right)\in \mathcal{U}\right\}$

is A-universal for ∃ ^ω Γ(E). First

${\exists }^{\omega }\mathcal{U}={\displaystyle \underset{n\in \mathbb{N}}{\cup }\left\{\left(y,x\right)\in A^{\omega }\times E|\left({\overline{y}}_n,x\right)\in \mathcal{U}\right\}}$

and since each map y → $\overline{y}$ _n is continuous, each set

$\left(y,x\right)\in A^{\omega }\times E|\left({\overline{y}}_n,x\right)\in \mathcal{U}\}$

belongs to Γ(A ^ω × E). Therefore ∃ ^ω $\mathcal{U}$ belongs to ∃ ^ω Γ(E). Furthermore, we have

$\begin{array}{c}{\exists }^{\omega }\Gamma \left(E\right)=\left\{{\displaystyle \underset{n\in \mathbb{N}}{\cup }{X}_n|X_n\in \Gamma \left(E\right)}\right\}=\left\{{\displaystyle \underset{n\in \mathbb{N}}{\cup }\mathcal{U}\left(y_n\right)|{\left(y_n\right)}_{n\in \mathbb{N}}\in {\left(A^{\omega }\right)}^{\omega }}\right\}\\ =\left\{{\displaystyle \underset{n\in \mathbb{N}}{\cup }\mathcal{U}\left({\overline{y}}_n\right)|y\in A^{\omega }}\right\}=\left\{{\displaystyle \underset{n\in \mathbb{N}}{\cup }\{x\in E|\left({\overline{y}}_n,x\right)\in \mathcal{U}|y\in A^{\omega }}\right\}\\ =\left\{\left\{x\in E|\exists n\in \mathbb{N}\left({\overline{y}}_n,x\right)\in \mathcal{U}\right\}|y\in A^{\omega }\right\}=\left\{{\exists }^{\omega }\mathcal{U}|\left(y\right)|y\in A^{\omega }\right\}\end{array}$

showing that ∃ ^ω $\mathcal{U}$ is universal for ∃ ^ω Γ(E).

We first establish a generic result on universal sets.

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Generalizations of Riemann Integration

Agamirza Bashirov , in Mathematical Analysis Fundamentals, 2014

Exercises

10.1

Generalize Exercise 9.1 to Riemann–Stieltjes integrals.

Hint: Elements of Cauchy criterion for Riemann–Stieltjes integrals was used in the proof of Theorem 10.5(b).

10.2

Prove Theorem 10.4.

10.3

Prove Theorem 10.5(a).

10.4

Prove Theorem 10.8.

10.5

Prove Theorem 10.9.

10.6

(Change of variable in the Riemann–Stieltjes integral) Let $(f\text{,}u)\in \mathit{RS}(a\text{,}b)$ and $g$ be a continuous bijection from $[a\text{,}b]$ on to $[c\text{,}d]$ . Prove that

${\int }_{g(a)}^{g(b)}f(t)\mathit{du}(t)={\int }_a^{b}f(g(x))\mathit{du}(g(x))\text{.}$

Hint: By Theorem 8.7(a), $g$ can be taken either strictly increasing or strictly decreasing.

10.7

(Continuity under the Riemann–Stieltjes integral) Let $f\in C([a\text{,}b]\times [c\text{,}d])$ , $u\in \mathit{BV}(a\text{,}b)$ , and

$F(y)={\int }_a^{b}f(x\text{,}y)\mathit{du}(x)\text{,}c\le y\le d\text{.}$

Show that $F\in C(c\text{,}d)$ .

Hint: Consult with the proof of Theorem 9.34.

10.8

(Interchange the order of differentiation and Riemann–Stieltjes integration) Let $f\in C([a\text{,}b]\times [c\text{,}d])$ and $u\in \mathit{BV}(a\text{,}b)$ , and define

$F(y)={\int }_a^{b}f(x\text{,}y)\mathit{du}(x)\text{,}c\le y\le d\text{.}$

Show that if $f_y^{\prime }\in C([a\text{,}b]\times [c\text{,}d])$ , then $F$ is differentiable on $[c\text{,}d]$ and

$F^{\prime }(y)={\int }_a^b{f}_y^{\prime }(x\text{,}y)\mathit{du}(x)\text{.}$

Hint: Consult with the proof of Theorem 9.35.

10.9

(Interchange the order of the Riemann–Stieltjes integrals) Let $f\in C([a\text{,}b]\times [c\text{,}d])$ , $u\in \mathit{BV}(a\text{,}b)$ , and $v\in \mathit{BV}(c\text{,}d)$ . Define the functions $F$ and $G$ by

$F(y)={\int }_a^{b}f(x\text{,}y)\mathit{du}(x)\text{,}c\le y\le d\text{,}$

and

$G(x)={\int }_c^{d}f(x\text{,}y)\mathit{dv}(y)\text{,}a\le x\le b\text{.}$

Show that $(F\text{,}v)\in \mathit{RS}(c\text{,}d)\text{,}(G\text{,}u)\in \mathit{RS}(a\text{,}b)$ , and

${\int }_c^{d}F(y)\mathit{dv}(y)={\int }_a^{b}G(x)\mathit{du}(x)\text{.}$

Hint: Consult with the proof of Theorem 9.36.

10.10

Prove that the sufficiency part of Theorem 10.26 does not hold for Riemann–Stieltjes integration in general.

Hint: Let

$f(x)=\left\{\begin{array}{cc}0\hfill & \text{if}a\le x\le c\text{,}\hfill \\ 1\hfill & \text{if}c<x\le b\text{,}\hfill \end{array}\right.\text{and}u(x)=\left\{\begin{array}{cc}0\hfill & \text{if}a\le x<c\text{,}\hfill \\ 1\hfill & \text{if}c\le x\le b\text{,}\hfill \end{array}\right.$

where $c\in (a\text{,}b)$ . Then

$S^{\ast }(f\text{,}u\text{,}\mathcal{P})=\left\{\begin{array}{cc}0\hfill & \text{if}c\in \mathcal{P}\text{,}\hfill \\ 1\hfill & \text{if}c\notin \mathcal{P}\text{,}\hfill \end{array}\right.\text{and}{S}_{\ast }(f\text{,}u\text{,}\mathcal{P})=0\text{,}$

implying $S^{\ast }(f\text{,}u)=S_{\ast }(f\text{,}u)=0$ . Conclude that $(f\text{,}u)\in \mathit{RS}(a\text{,}b)$ . Then show that the pair $(f\text{,}u)$ , as defined before, has the following property: for every $S\in \mathbb{R}$ and for every $\delta >0$ , there exists a partition $\mathcal{P}$ with $\parallel \mathcal{P}\parallel \le \delta$ and a selection of tags such that $\mid S(f\text{,}u\text{,}\mathcal{P})-S\mid \ge 1/2$ .

10.11

Prove that if $(f\text{,}u)\in \mathit{RS}(a\text{,}b)$ , where $u\in \mathit{BV}(a\text{,}b)$ , then $(f\text{,}{V}_u)\in \mathit{RS}(a\text{,}b)$ .

10.12

Prove that a functional in a Banach space is linear if it is continuous and homogenous.

10.13

For fixed $u\in \mathit{BV}(a\text{,}b)$ , define the functional

$F(f)={\int }_a^{b}u(x)\mathit{df}(x)\text{,}f\in C(a\text{,}b)\text{,}$

and for fixed $f\in C(a\text{,}b)$ , define

$G(u)={\int }_a^{b}u(x)\mathit{df}(x)\text{,}u\in \mathit{BV}(a\text{,}b)\text{.}$

Prove that $F$ and $G$ are continuous linear functionals.

Hint: Use Theorem 10.7.

10.14

Prove Theorem 10.33.

10.15

Prove Theorem 10.34.

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Quantum Process Tomography and Regression

Peter Wittek , in Quantum Machine Learning, 2014

13.3 Groups, Compact Lie Groups, and the Unitary Group

Compact groups are natural extensions of finite groups, and many properties of finite groups carry over. Of them, compact Lie groups are the best understood. We will need their properties later, so we introduce a series of definitions that are related to these structures.

A group G is a finite or infinite set, together with an operation ⋅ such that

(13.10) $\begin{array}{rr}\hfill g_1\cdot g_2\in G& \forall g_1,g_2\in G,\\ (g_1\cdot g_2)\cdot g_3=g_1\cdot (g_2\cdot g_3)& \forall g_1,g_2,g_3\in G,\\ \begin{array}{cc}\exists e\in G:& e\cdot g=g\cdot e=g\end{array}& \forall g\in G,\\ \begin{array}{cc}\forall g\in G& \exists h:\end{array}& g\cdot h=h\cdot g=e.\end{array}$

We often omit the symbol ⋅ for the operation, and simply write g ₁ g ₂ to represent the composition. The element e is the unit element of the group. The element h in this context is the inverse of g, and it is often denoted as g ⁻¹.

A topological group is a group G where the underlying set is also a topological space such that the group's operation and the group's inverse function are continuous functions with respect to the topology. The topological space of the group defines sets of neighborhoods for each group element that satisfy a set of axioms relating the elements and the neighborhoods. The neighborhoods are defined as subsets of the set of G, called open sets, satisfying these conditions.

A function f between topological spaces is called continuous if for all g ∈ G and all neighborhoods N of f (g) there is a neighborhood M of g such that f (M) ⊆ N . A special class of continuous functions is called homeomorphisms: these are continuous bijections for which the inverse function is also continuous. Topological groups introduce a sense of duality: we can perform the group operations on the elements of the set, and we can talk about continuous functions due to the topology.

A topological manifold is a topological space that is furthered characterized by having a structure which locally resembles real n-dimensional space. A topological space X is called locally Euclidean if there is a nonnegative integer n such that every point in X has a neighborhood which is homeomorphic to the Euclidean space ℝ ⁿ . A topological manifold is a locally Euclidean Hausdorff space—that is, distinct points have distinct neighborhoods in the space.

To do calculus, we need a further definition. A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition maps are all differentiable. Here an atlas is a collection of charts, where each chart is a linear space where the usual rules of calculus apply. The differentiable transitions between the charts ensure there is a global structure. A smooth manifold is a differentiable manifold for which all the transition maps have derivatives of all orders—that is, they are smooth.

The underlying set of a Lie group is also a finite-dimensional smooth manifold, and in which the group operations of multiplication and inversion are also smooth maps. Smoothness of the group multiplication means that it is a smooth mapping of the product manifold G × G to G.

A compact group is a topological group whose topology is compact. The intuitive view of compactness is that it generalizes closed and bounded subsets of Euclidean spaces. Formally, a topological space X is called compact if each of its open covers has a finite subcover—that is, for every collection {U _α } _α∈A of open subsets of X such that X = ⋃ _α∈A U _α there is a finite subset J of A such that X = ⋃ _i∈J U _i .

A compact Lie group has all the properties described so far, and it is a well- understood structure. An intuitive, albeit somewhat rough way to think about compact Lie groups is that they contain symmetries that form a bounded set.

To gain insight into these new definitions, we consider an example, the circle group, denoted by U(1), which is a one-dimensional compact Lie group. It is the unit circle on the complex plain with complex multiplication as the group operation:

(13.11) $U(1)=\{z\in ℂ:|z|=1\}.$

The notation U(1) refers to the interpretation that this group can also be viewed as 1 × 1 unitary matrices acting on the complex plane by rotation about the origin.

Complex multiplication and inversion are continuous functions on this set; hence, it is a topological group.

Furthermore, the circle is a one-dimensional topological manifold. As multiplication and inversion are analytic maps on the circle, it is a smooth manifold. The unit circle is a closed subset of the complex plane; hence, it is a compact group. The circle group is indeed a compact Lie group.

If we generalize this example further, the unitary group U(n) of degree n is the group of n × n unitary matrices, with the matrix multiplication as the group operation. It is a finite-dimensional compact Lie group.

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Sets and Extensions in the Twentieth Century

Akihiro Kanamori , in Handbook of the History of Logic, 2012

1.2 Continuum Hypothesis and transfinite numbers

By his next publication [1878] Cantor had shifted the weight to getting bijective correspondences, stipulating that two sets have the same power [Mächtigkeit] iff there is such a correspondence between them, and established that the reals

and the n-dimensional spaces

ⁿ all have the same power. Having made the initial breach in [1874] with a negative result about the lack of a bijective correspondence, Cantor secured the new ground with a positive investigation of the possibilities for having such correspondences. ¹² With "sequence" tied traditionally to countability through the indexing, Cantor used "correspondence [Beziehung]". Just as the discovery of the irrational numbers had led to one of the great achievements of Greek mathematics, Eudoxus's theory of geometrical proportions presented in Book V of Euclid's Elements and thematically antecedent to Dedekind's [1872] cuts, Cantor began his move toward a full-blown mathematical theory of the infinite.

Although holding the promise of a rewarding investigation Cantor did not come to any powers for infinite sets other than the two as set out in his [1874] proof. Cantor claimed at the end of [1878:257]:

Every infinite set of reals either is countable or has the power of the continuum.

This was the Continuum Hypothesis (CH) in the nascent context. The conjecture viewed as a primordial question would stimulate Cantor not only to approach the reals qua extensionalized continuum in an increasingly arithmetical fashion but also to grapple with fundamental questions of set existence. His triumphs across a new mathematical context would be like a brilliant light to entice others into the study of the infinite, but his inability to establish CH would also cast a long shadow. Set theory had its beginnings not as some abstract foundation for mathematics but rather as a setting for the articulation and solution of the Continuum Problem: to determine whether there are more than two powers embedded in the continuum.

In his magisterial Grundlagen [1883] Cantor developed the transfinite numbers [Anzahlen] and the key concept of well-ordering. A well-ordering of a set is a linear ordering of it according to which every non-empty subset has a least element. No longer was the infinitary indexing of his trigonometric series investigations mere contrivance. The "symbols of infinity" became autonomous and extended as the transfinite numbers, the emergence signified by the notational switch from the ∞ of potentiality to the ω of completion as the last letter of the Greek alphabet. With this the progression of transfinite numbers could be depicted:

$0,1,2,\dots \omega ,\omega +1,\omega +2,\dots ,\omega +\omega \left(=\omega \cdot 2\right),\dots ,{\omega }^2,\dots ,{\omega }^{\omega },\dots ,{\omega }^{{\omega }^{\omega }},\dots$

A corresponding transition from subsets of

ⁿ to a broader concept of set was signaled by the shift in terminology from "point-manifold [Punktmannigfaltigkeit]" to "set [Menge]". In this new setting well-orderings conveyed the sense of sequential counting and transfinite numbers served as standards for gauging well-orderings.

As Cantor pointed out, every linear ordering of a finite set is already a wellordering and all such orderings are isomorphic, so that the general sense is only brought out by infinite sets, for which there are non-isomorphic well-orderings. Cantor called the set of natural numbers the first number class (I) and the set of numbers whose predecessors are countable the second number class (II). Cantor conceived of (II) as being bounded above according to a limitation principle and showed that (II) itself is not countable. Proceeding upward, Cantor called the set of numbers whose predecessors are in bijective correspondence with (II) the third number class (III), and so forth. Cantor took a set to be of a higher power than another if they are not of the same power yet the latter is of the same power as a subset of the former. Cantor thus conceived of ever higher powers as represented by number classes and moreover took every power to be so represented. With this "free creation" of numbers, Cantor [1883:550] propounded a basic principle that was to drive the analysis of sets:

"It is always possible to bring any well-defined set into the form of a well-ordered set."

He regarded this as a "an especially remarkable law of thought which through its general validity is fundamental and rich in consequences." Sets are to be well-ordered, and thus they and their powers are to be gauged via the transfinite numbers of his structured conception of the infinite.

The well-ordering principle was consistent with Cantor's basic view in the Grundlagen that the finite and the transfinite are all of a piece and uniformly comprehendable in mathematics, ¹³ a view bolstered by his systematic development of the arithmetic of transfinite numbers seamlessly encompassing the finite numbers. Cantor also devoted several sections of the Grundlagen to a justificatory philosophy of the infinite, and while this metaphysics can be separated from the mathematical development, one concept was to suggest ultimate delimitations for set theory: Beyond the transfinite was the "Absolute", which Cantor eventually associated mathematically with the collection of all ordinal numbers and metaphysically with the transcendence of God. ¹⁴

The Continuum Problem was never far from this development and could in fact be seen as an underlying motivation. The transfinite numbers were to provide the framework for Cantor's two approaches to the problem, the approach through power and the more direct approach through definable sets of reals, these each to initiate vast research programs.

As for the approach through power, Cantor in the Grundlagen established that the second number class (II) is uncountable, yet any infinite subset of (II) is either countable or has the same power as (II). Hence, (II) has exactly the property that Cantor sought for the reals, and he had reduced CH to the positive assertion that the reals and (II) have the same power. The following in brief is Cantor's argument that (II) is uncountable:

Suppose that s is a (countable) sequence of members of (II), say with initial element a. Let a′ be a member of s, if any, such that a < a′; let a″ be a member of s, if any, such that a′ < a″; and so forth. Then however long this process continues, the supremum of these numbers, or its successor, is not a member of s.

This argument was reminiscent of his [1874] argument that the reals are uncountable and suggested a correlation of the reals through their fundamental sequence representation with the members of (II) through associated cofinal sequences. ¹⁵ However, despite several announcements Cantor could never develop a workable correlation, an emerging problem in retrospect being that he could not define a well-ordering of the reals.

As for the approach through definable sets of reals, this evolved directly from Cantor's work on trigonometric series, the "symbols of infinity" used in the analysis of the P′ operation transmuting to the transfinite numbers of the second number class (II). ¹⁶ In the Grundlagen Cantor studied P′ for uncountable P and defined the key concept of a perfect set of reals (non-empty, closed, and containing no isolated points). Incorporating an observation of Ivar Bendixson [1883], Cantor showed in the succeeding [1884] that any uncountable closed set of reals is the union of a perfect set and a countable set. For a set A of reals, A has the perfect set property iff A is countable or else has a perfect subset. Cantor had shown in particular that closed sets have the perfect set property.

Since Cantor [1884; 1884a] had been able to show that any perfect set has the power of the continuum, he had established that "CH holds for closed sets": every closed set either is countable or has the power of the continuum. Or from his new vantage point, he had reduced the Continuum Problem to determining whether there is a closed set of reals of the power of the second number class. He was unable to do so, but he had initiated a program for attacking the Continuum Problem that was to be vigorously pursued (cf. 2.3 and 2.5).

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SET THEORY FROM CANTOR TO COHEN

Akihiro Kanamori , in Philosophy of Mathematics, 2009

1.2 Continuum Hypothesis and Transfinite Numbers

By his next publication [1878] Cantor had shifted the weight to getting bijective correspondences, stipulating that two sets have the same power [Mächtigkeit] iff there is such a correspondence between them, and established that the reals ℝ and the n-dimensional spaces ℝ ⁿ all have the same power. Having made the initial breach in [1874] with a negative result about the lack of a bijective correspondence, Cantor secured the new ground with a positive investigation of the possibilities for having such correspondences. ¹² With "sequence" tied traditionally to countability through the indexing, Cantor used "correspondence [Beziehung]". Just as the discovery of the irrational numbers had led to one of the great achievements of Greek mathematics, Eudoxus's theory of geometrical proportions presented in Book V of Euclid's Elements and thematically antecedent to Dedekind's [1872] cuts, Cantor began his move toward a full-blown mathematical theory of the infinite.

Although holding the promise of a rewarding investigation Cantor did not come to any powers for infinite sets other than the two as set out in his [1874] proof. Cantor claimed at the end of [1878: 257]:

Every infinite set of reals either is countable or has the power of the continuum. This was the Continuum Hypothesis (CH) in the nascent context. The conjecture viewed as a primordial question would stimulate Cantor not only to approach the reals qua extensionalized continuum in an increasingly arithmetical fashion but also to grapple with fundamental questions of set existence. His triumphs across a new mathematical context would be like a brilliant light to entice others into the study of the infinite, but his inability to establish CH would also cast a long shadow. Set theory had its beginnings not as some abstract foundation for mathematics but rather as a setting for the articulation and solution of the Continuum Problem: to determine whether there are more than two powers embedded in the continuum.

In his magisterial Grundlagen [1883] Cantor developed the transfinite numbers [Anzahlen] and the key concept of well-ordering. A well-ordering of a set is a linear ordering of it according to which every non-empty subset has a least element. No longer was the infinitary indexing of his trigonometric series investigations mere contrivance. The "symbols of infinity" became autonomous and extended as the transfinite numbers, the emergence signified by the notational switch from the ∞ of potentiality to the ω of completion as the last letter of the Greek alphabet. With this the progression of transfinite numbers could be depicted:

$0,1,2,\dots \omega ,\omega +1,\omega +2,\dots ,\omega +\omega (=\omega \cdot 2),\dots ,{\omega }^2,\dots ,{\omega }^{\omega },\dots ,{\omega }^{{\omega }^{\omega }},\dots$

A corresponding transition from subsets of ℝ ⁿ to a broader concept of set was signaled by the shift in terminology from "point-manifold [Punktmannigfaltigkeit]" to "set [Menge]". In this new setting well-orderings conveyed the sense of sequential counting and transfinite numbers served as standards for gauging well-orderings.

As Cantor pointed out, every linear ordering of a finite set is already a well-ordering and all such orderings are isomorphic, so that the general sense is only brought out by infinite sets, for which there are non-isomorphic well-orderings. Cantor called the set of natural numbers the first number class (I) and the set of numbers whose predecessors are countable the second number class (II). Cantor conceived of (II) as being bounded above according to a limitation principle and showed that (II) itself is not countable. Proceeding upward, Cantor called the set of numbers whose predecessors are in bijective correspondence with (II) the third number class (III), and so forth. Cantor took a set to be of a higher power than another if they are not of the same power yet the latter is of the same power as a subset of the former. Cantor thus conceived of ever higher powers as represented by number classes and moreover took every power to be so represented. With this "free creation" of numbers, Cantor [1883:550] propounded a basic principle that was to drive the analysis of sets:

"It is always possible to bring any well-defined set into the form of a well-ordered set."

He regarded this as a "an especially remarkable law of thought which through its general validity is fundamental and rich in consequences." Sets are to be well-ordered, and thus they and their powers are to be gauged via the transfinite numbers of his structured conception of the infinite.

The well-ordering principle was consistent with Cantor's basic view in the Grundlagen that the finite and the transfinite are all of a piece and uniformly comprehendable in mathematics, ¹³ a view bolstered by his systematic development of the arithmetic of transfinite numbers seamlessly encompassing the finite numbers. Cantor also devoted several sections of the Grundlagen to a justificatory philosophy of the infinite, and while this metaphysics can be separated from the mathematical development, one concept was to suggest ultimate delimitations for set theory: Beyond the transfinite was the "Absolute", which Cantor eventually associated mathematically with the collection of all ordinal numbers and metaphysically with the transcendence of God. ¹⁴

The Continuum Problem was never far from this development and could in fact be seen as an underlying motivation. The transfinite numbers were to provide the framework for Cantor's two approaches to the problem, the approach through power and the more direct approach through definable sets of reals, these each to initiate vast research programs.

As for the approach through power, Cantor in the Grundlagen established that the second number class (II) is uncountable, yet any infinite subset of (II) is either countable or has the same power as (II). Hence, (II) has exactly the property that Cantor sought for the reals, and he had reduced CH to the positive assertion that the reals and (II) have the same power. The following in brief is Cantor's argument that (II) is uncountable:

Suppose that s is a (countable) sequence of members of (II), say with initial element a. Let a' be a member of s, if any, such that a < a'; let a" be a member of s, if any, such that a' < a"; and so forth. Then however long this process continues, the supremum of these numbers, or its successor, is not a member of s.

This argument was reminiscent of his [1874] argument that the reals are uncountable and suggested a correlation of the reals through their fundamental sequence representation with the members of (II) through associated cofinal sequences. ¹⁵ However, despite several announcements Cantor could never develop a workable correlation, an emerging problem in retrospect being that he could not define a well-ordering of the reals.

As for the approach through definable sets of reals, this evolved directly from Cantor's work on trigonometric series, the "symbols of infinity" used in the analysis of the P' operation transmuting to the transfinite numbers of the second number class (II). ¹⁶ In the Grundlagen Cantor studied P' for uncountable P and defined the key concept of a perfect set of reals (non-empty, closed, and containing no isolated points). Incorporating an observation of Ivar Bendixson [1883], Cantor showed in the succeeding [1884] that any uncountable closed set of reals is the union of a perfect set and a countable set. For a set A of reals, A has the perfect set property iff A is countable or else has a perfect subset. Cantor had shown in particular that closed sets have the perfect set property.

Since Cantor [1884; 1884a] had been able to show that any perfect set has the power of the continuum, he had established that "CH holds for closed sets": every closed set either is countable or has the power of the continuum. Or from his new vantage point, he had reduced the Continuum Problem to determining whether there is a closed set of reals of the power of the second number class. He was unable to do so, but he had initiated a program for attacking the Continuum Problem that was to be vigorously pursued (cf. 2.3 and 2.5).

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William Wistar Comfort (1933-2016): In Memoriam

Salvador Hernández , ... F. Javier Trigos-Arrieta , in Topology and its Applications, 2019

5 The Bohr compactification of a LCA group modulo a metrizable subgroup [28]

If N is a closed subgroup of bG, the authors of [28] say thay N preserves compactness if a subspace A of G satisfies that $\varphi [b[A]]$ is compact in $bG/N$ if and only if $A(b^{-1}[G^{+}\cap N])$ is compact in G, where $\varphi :bG⟶bG/N$ denotes the natural quotient map. When the closed metrizable subgroups of bG preserve compactness, we say that G strongly respects compactness. In [28] it is shown that LCAGs strongly respect compactness. This result generalizes Glicksberg's theorem [47] by taking $N:=\{0_{bG}\}$ above. In the same article, the authors show that the result can fail if N is not metrizable, even if $N\cap G^{+}=\{0_{G^{+}}\}$ ; but they also show that the result may still hold for non-metrizable closed subgroups N of G.

The case G discrete turned out to be more difficult than expected and a misleading and most likely incorrect proof was printed in [28]. The first author and Jorge Galindo noticed the mistake and informed the authors of [28] about it. Eventually, a sound proof of the case G discrete was found and subsequently published as a correction in [29].

The paper [28] then discusses the interplay between k-topologies and the Bohr topology. Recall that a k-space X is one in which every set U satisfying that $U\cap K$ is an open subset of every compact subspace K of X is itself open in X. If Y is a space, there exists a k-space X and a continuous bijection $f:X⟶Y$ such that K compact in $Y⟹f^{-1}[K]$ compact in X, and one shows that, up to a homeomorphism such X is unique. We then write $kY=X$ . W. F. LaMartin [61] (2.1) showed that $k({\mathbb{R}}^{\alpha })$ is not a topological group whenever $\alpha >{\mathrm{\aleph }}_0$ , hence the k-ification of a topological group need not be a topological group. Glicksberg's theorem [47] implies then that if G is a LCAG, then $G=k{G}^{+}$ . The main result of [28] yields a totally bounded group H with $G:=kH$ being a locally compact Abelian group, yet $H\ne G^{+}$ . The characterization of those totally bounded groups H such that $G:=kH$ is a locally compact Abelian group, and $H=G^{+}$ , was posed as Question 4.3, studied by J. Galindo in [43], and recently solved by the first and last authors in the paper [53].

The rest of [28] consists of constructing examples of LCAGs G and closed, non-metrizable subgroups N of bG preserving compactness. Paramount for this is lemma 3.11, a very interesting result in itself: Assume that G is a discrete Abelian group, N a closed subgroup of bG with $N\cap G=\{0\}$ , and λ the Haar measure of $\hat{G}$ . Let $\mathbb{A}(\widehat{bG},N):=\{\chi \in \widehat{bG}:\chi [N]=\{0\}\}$ , i.e., $\mathbb{A}(\widehat{bG},N)$ is the annihilator of N in $\widehat{bG}$ . Suppose that either $\{{\chi }_{|G}:\chi \in \mathbb{A}(\widehat{bG},N)\}$ is not λ-measurable in $\hat{G}$ , or $\lambda (\{{\chi }_{|G}:\chi \in \mathbb{A}(\widehat{bG},N)\})>0$ . Denote by $\varphi :G⟶bG/N$ the natural map. If G is countable or $\varphi [H]$ is closed in $bG/N$ , whenever H is a subgroup of G, then N preserves compactness. So, here it is a connection between preserving compactness and the Haar measure of the character group, that has been exploited by several mathematicians. In another direction, the class of groups that strongly preserve compactness has been considered by different authors for Abelian and non-Abelian groups. We refer to [39], [45] and the references therein.

The article [28] concludes with a number of questions, some of them already answered but others still open. We already mentioned the solution to Question 4.3, and in the contribution [53] the first and last authors show that there is a MAP group G not strongly respecting compactness, yet having all closed metrizable subgroups N of bG with $N\cap G^{+}=\{0\}$ , preserving compactness, answering hence Question 4.1 of [28].

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