Algebraic General Topology. Volume 1 addons Victor Porton Email address: [email protected] URL: http://www.mathematics21.org 2000 Mathematics Subject Classification. 54J05, 54A05, 54D99, 54E05, 54E15, 54E17, 54E99 Key words and phrases. algebraic general topology, quasi-uniform spaces, generalizations of proximity spaces, generalizations of nearness spaces, generalizations of uniform spaces Abstract. This file contains future addons for the free e-book “Algebraic General Topology. Volume 1”, which are yet not enough ripe to be included into the book. Contents Chapter 1. About this document5 Chapter 2. Applications of algebraic general topology6 1. “Hybrid” objects6 2. A way to construct directed topological spaces6 3. Some inequalities8 4. Continuity9 5. A way to construct directed topological spaces 12 6. Integral curves 12 Chapter 3. More on generalized limit 16 1. Hausdorff funcoids 16 2. Restoring functions from limit 16 Chapter 4. Extending Galois connections between funcoids and reloids 18 Chapter 5. Boolean funcoids 20 1. One-element boolean lattice 20 2. Two-element boolean lattice 20 3. Finite boolean lattices 21 4. About infinite case 21 Chapter 6. Interior funcoids 23 Chapter 7. Filterization of pointfree funcoids 25 Chapter 8. Systems of sides 26 1. More on Galois connections 26 2. Definition 27 3. Concrete examples of sides 28 4. Product 30 5. Negative results 31 6. Dagger systems of sides 31 Chapter 9. Backward Funcoids 32 Chapter 10. Quasi-atoms 33 Chapter 11. Cauchy Filters on Reloids 34 1. Preface 34 2. Low spaces 34 3. Almost sub-join-semilattices 35 4. Cauchy spaces 35 5. Relationships with symmetric reloids 36 6. Lattices of low spaces 37 7. Up-complete low spaces 41 3 CONTENTS 4 8. More on Cauchy filters 42 9. Maximal Cauchy filters 43 10. Cauchy continuous functions 44 11. Cauchy-complete reloids 44 12. Totally bounded 44 13. Totally bounded funcoids 45 14. On principal low spaces 45 15. Rest 45 Chapter 12. Funcoidal groups 47 1. On “Each regular paratopological group is completely regular” article 48 Chapter 13. Micronization 53 Chapter 14. More on connectedness 54 1. For topological spaces 54 Chapter 15. Relationships are pointfree funcoids 59 Chapter 16. Manifolds and surfaces 60 1. Sides of a surface 60 2. Special points 60 Bibliography 63 CHAPTER 1 About this document This file contains future addons for the free e-book “Algebraic General Topol- ogy. Volume 1”, which are yet not enough ripe to be included into the book. Theorem (including propositions, conjectures, etc.) numbers in this document start from the last theorem number in the book plus one. Theorems references inside this document are hyperlinked, but references to theorems in the book are not hyperlinked (because PDF viewer Okular 0.20.2 does not support Backward button after clicking a cross-document reference, and thus I want to avoid clicking such links). 5 CHAPTER 2 Applications of algebraic general topology 1. “Hybrid” objects Algebraic general topology allows to construct “hybrid” objects of “continuous” (as topological spaces) and discrete (as graphs). Consider for example D t T where D is a digraph and T is a topological space. The n-th power (D tT )n yields an expression with 2n terms. So treating D tT as one object (what becomes possible using algebraic general topology) rather than the join of two objects may have an exponential benefit for simplicity of formulas. 2. A way to construct directed topological spaces 2.1. Some notation. I use E and ι notations from volume-2.pdf. FiXme: Reorder document fragments to describe it before use. I remind that f|X = f ◦ idX for binary relations, funcoids, and reloid. X −1 f kX = f ◦ (E ) . −1 fX = idX ◦f ◦ idX . As proved in volume-2.pdf, the following are bijections and moreover isomor- phisms (for R being either funcoids or reloids or binary relations): n o ◦ (f|X ,fkX ) 1 . f∈R ; n o ◦ (fX,ιX f) 2 . f∈R . As easily follows from these isomorphisms and theorem 1293: Proposition 2064. For funcoids, reloids, and binary relations: ◦ 1 . f ∈ C(µ, ν) ⇒ f kA∈ C(ιAµ, ν); ◦ 0 0 2 . f ∈ C (µ, ν) ⇒ f kA∈ C (ιAµ, ν); ◦ 00 00 3 . f ∈ C (µ, ν) ⇒ f kA∈ C (ιAµ, ν). 2.2. Directed line and directed intervals. Let A be a poset. We will denote A = A ∪ {−∞, +∞} the poset with two added elements −∞ and +∞, such that +∞ is strictly greater than every element of A and −∞ is strictly less. FiXme: Generalize from R to a wider class of posets. Definition 2065. For an element a of a poset A ◦ n x∈A o 1 . J≥(a) = x≥a ; ◦ x∈A 2 . J>(a) = x>a ; ◦ n x∈A o 3 . J≤(a) = x≤a ; ◦ x∈A 4 . J<(a) = x<a ; ◦ n x∈A o 5 . J6=(a) = x6=a . Definition 2066. Let a be an element of a poset A. n o 1◦. ∆(a) = F ]x;y[ ; d x,y∈A,x<a∧y>a n o 2◦. ∆ (a) = F [a;y[ ; ≥ d y∈A,y>a 6 2. A WAY TO CONSTRUCT DIRECTED TOPOLOGICAL SPACES 7 n o 3◦. ∆ (a) = F ]a;y[ ; > d y∈A,x<a∧y>a n o 4◦. ∆ (a) = F ]x;a] ; ≤ d x∈A,x<a n o 5◦. ∆ (a) = F ]x;a[ ; < d x∈A,x<a ◦ 6 . ∆6=(a) = ∆(a) \{a}. Obvious 2067. ◦ F 1 . ∆≥(a) = ∆(a) u @J≥(a); ◦ F 2 . ∆>(a) = ∆(a) u @J>(a); ◦ F 3 . ∆≤(a) = ∆(a) u @J≤(a); ◦ F 4 . ∆<(a) = ∆(a) u @J<(a); ◦ F 5 . ∆6=(a) = ∆(a) u @J6=(a). Definition 2068. Given a partial order A and x ∈ A, the following defines complete funcoids: 1◦. h|A|i∗{x} = ∆(x); ◦ ∗ 2 . h|A|≥i {x} = ∆≥(x); ◦ ∗ 3 . h|A|>i {x} = ∆>(x); ◦ ∗ 4 . h|A|≤i {x} = ∆≤(x); ◦ ∗ 5 . h|A|<i {x} = ∆<(x); ◦ ∗ 6 . h|A|6=i {x} = ∆6=(x). Proposition 2069. The complete funcoid corresponding to the order topol- ogy1 is equal to |A|. Proof. Because every open set is a finite union of open intervals, the com- plete funcoid f corresponding to the order topology is described by the formula: n o hfi∗{x} = F ]a;b[ = ∆(x) = h|A|i∗{x}. Thus f = |A|. d a,b∈A,a<x∧b>x Exercise 2070. Show that |A|≥ (in general) is not the same as “right order topology”2. Proposition 2071. D E∗ n o 1◦. |A|−1 @X = @ a∈A ; ≥ ∀y∈A:(y>a⇒X∩[a;y[6=∅) ∗ n o 2◦. |A|−1 @X = @ a∈A ; > ∀y∈A:(y>a⇒X∩]a;y[6=∅) D E∗ n o 3◦. |A|−1 @X = @ a∈A ; ≤ ∀x∈A:(x<a⇒X∩]x;a]6=∅) ∗ n o 4◦. |A|−1 @X = @ a∈A . < ∀x∈A:(x<a⇒X∩]x;a[6=∅) ∗ ∗ D −1E D −1E ∗ Proof. a ∈ |A|≥ @X ⇔ @{a} 6 |A|≥ @X ⇔ h|A|≥i @{a} 6 @X ⇔ ∆≥(a) 6 @X ⇔ ∀y ∈ A :(y > a ⇒ X ∩ [a; y[6= ∅). −1 ∗ −1 ∗ ∗ a ∈ |A|> @X ⇔ @{a} 6 |A|> @X ⇔ h|A|>i @{a} 6 @X ⇔ ∆>(a) 6 @X ⇔ ∀y ∈ A :(y > a ⇒ X∩]a; y[6= ∅). The rest follows from duality. Remark 2072. On trivial ultrafilters these obviously agree: ◦ ∗ ∗ 1 . h|R|≥i {x} = h|R|u ≥i {x}; ◦ ∗ ∗ 2 . h|R|>i {x} = h|R|u >i {x}; ◦ ∗ ∗ 3 . h|R|≤i {x} = h|R|u ≤i {x}; ◦ ∗ ∗ 4 . h|R|<i {x} = h|R|u <i {x}. 1See Wikipedia for a definition of “Order topology”. 2See Wikipedia 3. SOME INEQUALITIES 8 Corollary 2073. ◦ 1 . |R|≥ = Compl(|R|u ≥); ◦ 2 . |R|> = Compl(|R|u >); ◦ 3 . |R|≤ = Compl(|R|u ≤); ◦ 4 . |R|< = Compl(|R|u <). Obvious 2074. FiXme: also what is the values of \ operation ◦ 1 . |R|≥ = |R|> t 1; ◦ 2 . |R|≤ = |R|< t 1. 3. Some inequalities FiXme: Define the ultrafilter “at the left” and “at the right” of a real number. Also define “convergent ultrafilter”. Denote ∆+∞ = ]x; +∞[ and ∆−∞ = ] − ∞; x[. dx∈R dx∈R The following proposition calculates h≥ix and h>ix for all kinds of ultrafilters on R: Proposition 2075. 1◦. h≥i{α} = [α; +∞[ and h>i{α} =]α; +∞[. 2◦. h≥ix = h>ix =]α; +∞[ for ultrafilter x at the right of a number α. ◦ 3 . h≥ix = h>ix = ∆<(α)t[α; +∞[= ∆≤(α)t]α; +∞[ for ultrafilter x at the left of a number α. ◦ 4 . h≥ix = h>ix = ∆+∞ for ultrafilter x at positive infinity. 5◦. h≥ix = h>ix = R for ultrafilter x at negative infinity. Proof. 1◦. Obvious. 2◦. F F h≥ix = l h≥i(Xu]α; +∞[) = l ]α; +∞[=]α; +∞[; X∈up x X∈up x F F h>ix = l h>i(Xu]α; +∞[) = l ]α; +∞[=]α; +∞[. X∈up x X∈up x ◦ 3 . ∆<(α) t [α; +∞[= ∆≤(α)t]α; +∞[ is obvious. F F h>ix = l h>iX w l (∆<(α)t]α; +∞[) = ∆<(α)t]α; +∞[ X∈up x X∈up x but h≥ix v ∆<(α) t [α; +∞[ is obvious. It remains to take into account that h>ix v h≥ix. ◦ F F 4 . h≥ix = dX∈up xh≥iX = dX∈up x,inf X∈X h≥i(Xu]α; +∞[) = F F F dX∈up x[inf X; +∞[= dx>α[x; +∞[= ∆+∞; h>ix = dX∈up xh>iX = F F F dX∈up x,inf X∈X h>i(Xu]α; +∞[) = dX∈up x] inf X; +∞[= dx>α[x; +∞[= ∆+∞. ◦ F 5 . h≥ix w h>ix = dX∈up xh>iX but h>iX =] − ∞; +∞[ for X ∈ up x because X has arbitrarily small elements. Lemma 2076. h|R|ix v h>ix = h≥ix for every nontrivial ultrafilter x.