Non-deterministic

N o n - D e t e r m i n i s t i c A na ly s i s and A p p l i c at i o n s

universidade estadual de campinas Reitor José Tadeu Jorge Coordenador Geral da Universidade Alvaro Penteado Crósta

Conselho Editorial Presidente Eduardo Guimarães Elinton Adami Chaim – Esdras Rodrigues Silva Guita Grin Debert – Julio Cesar Hadler Neto Luiz Francisco Dias – Marco Aurélio Cremasco Ricardo Antunes – Sedi Hirano Unicamp Ano 50 Comissão Editorial Itala M. Loffredo D’Ottaviano Eduardo Guimarães

Rubens G. Lintz

N o n - D e t e r m i n i s t i c A na ly s i s and A p p l i c at i o n s

ficha catalográfica elaborada pelo sistema de bibliotecas da unicamp diretoria de tratamento da informação L658n

Lintz, Rubens Gouveia, 1930Non-deterministic analysis and applications / Rubens G. Lintz – Campinas, SP: Editora da Unicamp, 2015.

1.Topologia. 2. Topologia algébrica. 3. Análise numérica. 4. Física teórica. 5. Sistemas dinâmicos diferenciais. I. Título.

cdd 514

514.2

519.4

530 515.352 e-isbn 978-85-268-1315-1

Índices para catálogo sistemático:

1. Topologia 2. Topologia algébrica 3. Análise numérica 4. Física teórica 5. Sistemas dinâmicos diferenciais

Copyright © by Rubens G. Lintz Copyright © 2015 by Editora da Unicamp

Direitos reservados e protegidos pela Lei 9.610 de 19.2.1998. É proibida a reprodução total ou parcial sem autorização, por escrito, dos detentores dos direitos.

Printed in Brazil.

Foi feito o depósito legal.

Direitos reservados à Editora da Unicamp Rua Caio Graco prado, 50 – Campus Unicamp cep 13083-892 – Campinas – sp – Brasil Tel./Fax: (19) 3521-7718/7728 www.editora.unicamp.br – vendas@editora.unicamp.br

514 514.2 519.4 530 515.352

To the memory of my beloved wife, Valderes

Preface

In this book we intend to expose the fundamental notions of nondeterministic analysis and its application.

The subject started in 1964 based on considerations to be explained in §I and developed since then through the work of several students and collaborators reaching a stage which demands for a book containing the basic facts known up to now concerning nondeterministic analysis and its application collected under a common roof unifying definitions and notations scattered over many papers, Ph.D. theses and other publications. In few words, we have tried our best for this book to be a general reference for future workers in this area.

The book contains two parts: Part I, dedicated to the study of nondeterministic analysis as complete as possible including all main results known up to now; Part II dedicated to the applications of nondeterministic analysis to several areas of mathematics and physics.

Each part is divided into sections denoted by §I, §II, etc. and each section is divided into paragraphs denoted by numbers 1,2, and subnumbers 1.1, 2.1, etc. The several theorems, propositions, lemmas, etc. are numbered in each section and numbering begins afresh in new sections.

The references are indicated by the author’s name followed by the work numbered [1], [2], etc. like (A. Jensen [1]) and are quoted at the end of the book for both parts.

Finally, I wish to acknowledge Mrs. Debbie Iscoe whose expertise in computer work was able to transform a huge mass of almost illegible manuscripts into a beautiful text acceptable for publication.

Contents Part I – Fundamental Concepts

1

Chapter I – n-Functions

3

§I – Generalities and Fundamental Notions . . . . . . . . . . .

5

§II – n-Functions and Usual Functions . . . . . . . . . . . . . 58 §III – n-Distributions and Related Subjects . . . . . . . . . . 91 §IV – Particular Classes of n-Functions . . . . . . . . . . . . . 123 §V – Categorical Approach to n-Functions . . . . . . . . . . . 134 Chapter II – Derivatives and Integral of n-Functions

151

§I – Gauss Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 153 §II – Derivatives of n-Functions . . . . . . . . . . . . . . . . . 195 §III – Integral of n-Functions . . . . . . . . . . . . . . . . . . 219 Part II – Non-Deterministic Analysis and Applications

253

Chapter I – Applications to Topology

255

§I – Cech Homology . . . . . . . . . . . . . . . . . . . . . . . 257 §II – Non-Deterministic Analysis and General Topology . . . . 275 §III – Non-Deterministic Dynamical Systems . . . . . . . . . . 297 Chapter II – Applications to Analysis

ix

351

§I – Limits and Related Areas . . . . . . . . . . . . . . . . . . 353 §II – n-Functions on Measure Spaces . . . . . . . . . . . . . . 364 §III – Differentiable Manifolds . . . . . . . . . . . . . . . . . . 370 Chapter III – Applications to Physics

385

§I – Physical Systems . . . . . . . . . . . . . . . . . . . . . . . 387 §II – Space, Time and Related Subjects . . . . . . . . . . . . . 408 Appendix and References

437

Appendix

439

§I – Space of n-Functions

. . . . . . . . . . . . . . . . . . . . 439

§II – Anti-Continuity of n-Functions . . . . . . . . . . . . . . 446 §III – Some Special Questions on n-Functions and Derivatives 448 References

450

x

Part I Fundamental Concepts

Chapter I n-Functions

§I Generalities and Fundamental Notions –1– 1.1 We introduce here the main notations to be used in this book together with a recollection of the fundamentals from set theory and topology, whose background the reader is supposed to be familiar.

All topological spaces are supposed to satisfy the T1 -axiom of separation, namely, points or elements of a topological space are closed subsets of that space, when considered as subspaces having only one element or singletons.

We shall use the following notations: the closure of a subset A ⊂ X o is denoted by A its interior by A and its boundary by bdA. The complement of A ⊂ X in X, namely, X − A will be denoted by ∁A and when several spaces are involved we write ∁ A to avoid misunderstanding.

X

If A ⊂ B ⊂ X in the topology relative to B, namely, that topology induced by X, we denote the boundary of A in B with bdB A, which in general is different from (bdA) ∩ B. Also the interior and the closure (0)

B

of A in B will be denoted by AB and A , respectively.

Given two set A and B, their symmetric difference, denoted by A∆B, is the set (A − B) ∪ (B − A).

A covering of a topological space A is a collection of subsets of 1.2 X, whose union is X. If all sets of this collection are open subsets of X we say that we have an open covering of X, if they are closed subsets of X we talk about closed covering of X and so on. If X happens to be a subset of another space E, namely X ⊂ E then a covering of X in E is a collection of subsets of E whose union contains X and when no confusion is possible we just say covering of X to mean a covering of X in E. In this book most covering will be made up of open sets and then we just say covering instead of open covering.

The set of all coverings of X will be denoted by Cov(X) either for X itself or if X ⊂ E. When it becomes necessary we use the notation CovE (X) to indicate the set of all coverings of X by open sets in E. 5

Let σ ∈ Cov(E) and X ⊂ E. We denote by σ ∩ X the set σ ∩ X = {A ∩ X : A ∈ σ} which is a covering of X by open sets in the relative topology of X in E. Call σ ∧ X, for σ ∈ Cov(E) and X ⊂ E, the set σ ∧ X = {A ∈ σ : A ∩ X 6= ∅}, which belongs also to CovE (X). Observe that the definition of covering of X in E does not exclude sets of E which do not intersect X. These sets are sometimes useless and then we only consider σ ∧X as coverings of X in E, for σ ∈ Cov(E).

Given σ, τ ∈ Cov(X), we say that τ refines σ, denoted τ > σ, if any B ∈ τ is contained in some A ∈ σ. Observe that this definition makes sense for arbitrary coverings of X, not necessarily open coverings.

If V is a family of covering of E, X ⊂ E we shall always suppose that V is ordered by refinements, namely the relation > define and ordering in V , usually called quasi-ordering. Let us introduce the notations V ∩ X = {σ ∩ X : σ ∈ V } V ∧ X = {σ ∧ X : σ ∈ V }.

Let σ, τ be coverings of X. We denote by σ ∧ τ the covering of X defined by: σ ∧ τ = {A ∩ B : A ∈ σ, B ∈ τ }.

Let us call σ ∨ τ = {A ∪ B : A ∈ σ or B ∈ τ }.

Do not confuse σ ∧ τ with σ ∩ τ , which is just the set intersection of σ and τ , namely, sets which belong both to σ and τ or σ ∨ τ with σ ∪ τ.

Let Γ ∈ CovX and A, B ⊂ X open sets, we say that A and B are Γ-equal, denoted by Γ A = B, if 6

1. A ∩ B = ∅ ⇒ A ∪ B ⊂ M ∈ Γ.

2. A ∩ B 6= ∅ ⇒ A∆B ⊂ M1 ∪ M2 with M1 , M2 ∈ Γ, A − B ⊂ M1 , B − A ⊂ M2 .

Let U and V be two families of coverings of X. We may say that U is cofinal in V if U ⊂ V and for any σ ∈ V there is σ 1 ∈ U such that σ 1 > σ. We say that U is cofinal in V relative to X if U ∩ X is cofinal in V ∩ X in the relative topology induced in X by E, when X ⊂ E. In this case we say that σ 1 refines σ relative to X if σ 1 ∩ X > σ ∩ X. When V = Cov(X) we just say that U is cofinal in X. We say that V is directed by refinements if given two σ, σ 1 ∈ V , there is τ ∈ V such that τ > σ and τ > σ 1 . Looking to V ∩ X we say that it is directed by refinements if for any σ, σ 1 ∈ V there is τ ∈ V such that τ ∩ X > σ ∩ X and τ ∩ X > σ 1 ∩ X. All definitions above can be extended “mutatis mutandis” to arbitrary collections of open sets in X or in E ⊃ X, not necessarily coverings of X. If in the definition of U cofinal in V we drop the assumption U ⊂ V we say that U is cofinal to V namely, we use the preposition in, when U ⊂ V and to, when not necessarily U ⊂ V .

Let Γ ∈ Cov(X). We call star of a set A ∈ Γ, denoted by St(A, Γ) the union of all sets B ∈ Γ such that B ∩ A 6= ∅.

We say that Λ ∈ Cov(X) is a star refinement of Γ if the star of any set of Λ is contained in some set of Γ.

For general properties of star refinements and connected subjects we recommend (Engelking, R. [1]).

Suppose we have now a space X which is the Cartesian product of spaces Xi , i = 1, 2, . . . , n, finite, with coverings σi of Xi . Let σ be the covering of X given by σ = {A1 × A2 × · · · × An : Ai ∈ σn , i = 1, 2, . . . , n} denoted by σ = σ1 ⊗ σ2 ⊗ σ3 ⊗ · · · ⊗ σn = 7

n O i=1

σi .

If Vi is a family of coverings of Xi , we call V the family of coverings of X made up of all σ defined above and denoted by V = V1 ⊗ V2 ⊗ · · · ⊗ Vn =

n O

Vi .

i=1

Let now X be the Cartesian multiplication of an arbitrary family of spaces Xi with the product topology, Y X= Xi i∈I

where I is an arbitrary index set for {Xi }, i ∈ I. If Vi is a family of coverings σi of Xi , i ∈ I we call O σ= σi i∈I

the covering of X defined as follows: for each J ⊂ I finite and nonempty consider Y A= Ai i∈I

where for i ∈ J, we have Ai ∈ σi and for i 6∈ J, Ai = Xi .

The σ as considered above is made up of all such A. Denote by O V = Vi . i∈I

The family of all σ defined as above, which is a family of coverings of X, called the Cartesian product of the families Vi , i ∈ I. 1.3

For later use we shall prove now a few lemmas.

Lemma A. Let X be the Cartesian product of a family {Xi }, i ∈ I, of spaces, as introduced above and Vi the families of coverings σi of Xi , i ∈ I, with V the Cartesian product of Vi , namely, O V = Vi . i∈I

Then τ > σ for τ, σ ∈ V if and only if for each i ∈ I we have τi > σi .

8

Proof.

Let τ=

O

τi

O

σi

i∈I

and σ=

i∈I

and assume that for each i ∈ I, τi > σi , and let Y B= Bi ∈ τ. i∈I

By the definition of B, there is a finite set J B ⊂ I such that, for i ∈ J B , Bi ∈ τi and for i 6∈ J , Bi = Xi . Let A ∈ σ with Y A= Ai ∈ σ i∈I

and let J A ⊂ I be a finite set such that, Ai ∈ σi for i ∈ J A and Ai = Xi for i 6∈ J A and such that Bi ⊂ Ai , i ∈ I. Clearly, J A ⊂ J B and thus B ⊂ A, namely τ > σ.

Suppose now that τ > σ and for an arbitrary i ∈ I let Bi ∈ τi and Y B= Bi ∈ τ. i∈I

By the definition of B we have that, if B ⊂ A for some A ∈ σ, then for each i ∈ J B we must have some Ai ∈ σi such that Bi ⊂ Ai and this implies that τi > σi . q.e.d.

Our next lemma is concerned with locally finite coverings, namely, a covering σ of X such that each x ∈ X has a neighborhood intersecting at most a finite number of elements of σ.

Lemma B. Let Γ be a locally finite covering of a space X whose elements are closure of open sets, namely if F ∈ Γ then there is A ⊂ X open such that F = A. Furthermore let us assume that for any o o F1 , F2 ∈ Γ we have F 1 ∩ F 2 = ∅.

Let x ∈ X arbitrary and call Γ(x) the union of all sets of Γ containing x. Then o x ∈ Γ(x).

9

Let, considering that Γ is locally finite,

Proof.

Γ(x) =

n [

x ∈ Fi .

Fi ,

i=1 o

If x ∈ Fi for some i, there is nothing to prove. Otherwise x ∈ bd Fi for all i, 1 ≤ i ≤ n. Suppose that x ∈ bd Γ(x), what implies that for any neighborhood V (x) of x there are points y ∈ V (x) with y 6∈ Γ(x). As Γ is locally finite we can assume that V (x) is selected in such a way that only finitely many set of Γ, say F1′ , . . . , Fk′ , intersect V (x) and are not contained in Γ(x). This implies that for an arbitrary neighborhood W (x) of x we must have W (x) ∩

k [

j=1

Fj′ 6= ∅

and as all Fj′ are closed we must have x∈

k [

Fj′

j=1

and consequently at least one of the Fj′ must be contained in Γ(x), or more precisely, for some j we must have Fj′ = Fi for some 1 ≤ i ≤ n, what is impossible. This completes the proof.

Due to Lemma B we introduce the following concept: for any covering Γ of X in the conditions of the lemma we call σ(Γ) the open covering o

whose open sets are Γ(x) for x ∈ X, the associated open covering to Γ.

Lemma C. Let X be the Cartesian multiplication of a family of spaces {Xi }, i ∈ I and Vi a family of open coverings σi of Xi for each i ∈ I. Then O V = Vi i∈I

is cofinal in Cov(X) if and only if for each i ∈ I, Vi is cofinal in Cov(Xi ).

Proof.

It is an easy consequence of Lemma A.

Lemma D. Let V be a family of open coverings of X, cofinal in Cov(X). Then for any open set B in X and any x ∈ B there is σ ∈ V and A ∈ σ such that x ∈ A ⊂ B. 10