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A point-free relation-algebraic approach to general topology - - PowerPoint PPT Presentation

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A point-free relation-algebraic approach to general topology Gunther Schmidt Fakult at f ur Informatik, Universit at der Bundeswehr M unchen Gunther.Schmidt@unibw.de May 1, 2014 Contents 1. Motivation my early topology 2.

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SLIDE 1

A point-free relation-algebraic approach to general topology

Gunther Schmidt

Fakult¨ at f¨ ur Informatik, Universit¨ at der Bundeswehr M¨ unchen Gunther.Schmidt@unibw.de

May 1, 2014

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Contents

  • 1. Motivation — my early topology
  • 2. Topology

3. Interlude on prerequisites

  • 4. Cryptomorphy of topology concepts
  • 5. Continuity

6. Interlude on structure comparison 7. Interlude on the existential and inverse image

  • 8. Relating continuity with the inverse image
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  • Triangulation of the projective plane
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  • Triangulation of the Csaszar polynomial
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  • Triangulation of the Csaszar torus
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SLIDE 6
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SLIDE 7
  • 5

6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

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SLIDE 8

Contents

  • 1. Motivation — my early topology
  • 2. Topology

3. Interlude on prerequisites

  • 4. Cryptomorphy of topology concepts
  • 5. Continuity

6. Interlude on structure comparison 7. Interlude on the existential and inverse image

  • 8. Relating continuity with the inverse image
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SLIDE 9

Topology — how it emerged

Leibniz: “geometria situs” the French: “g´ eom´ etrie de position”

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Topology — how it emerged

Leibniz: “geometria situs” the French: “g´ eom´ etrie de position” Johann Benedict Listing 1847: “Topologie” Karl von Staudt 1848: “Geometrie der Lage”

  • thers afterwards:

“analysis situs”

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SLIDE 11

Topology — how it emerged

Leibniz: “geometria situs” the French: “g´ eom´ etrie de position” Johann Benedict Listing 1847: “Topologie” Karl von Staudt 1848: “Geometrie der Lage”

  • thers afterwards:

“analysis situs” Definable via neighborhoods, open sets, open kernel, closed sets, etc.

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SLIDE 12

Topology — how it emerged

Leibniz: “geometria situs” the French: “g´ eom´ etrie de position” Johann Benedict Listing 1847: “Topologie” Karl von Staudt 1848: “Geometrie der Lage”

  • thers afterwards:

“analysis situs” Definable via neighborhoods, open sets, open kernel, closed sets, etc. Early in the twentieth century, topology has split into general

  • r point set theory, mainly invented by Georg Cantor and later

developed further by Felix Hausdorff, and what we today call algebraic topology, elaborated as Alexander Grothendieck’s cathedral.

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Topology as defined by Felix Hausdorff

A set X endowed with a system U(p) of subsets for every p 2 X is called a topological structure, provided

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SLIDE 14

Topology as defined by Felix Hausdorff

A set X endowed with a system U(p) of subsets for every p 2 X is called a topological structure, provided i) p 2 U for every neighborhood U 2 U(p)

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SLIDE 15

Topology as defined by Felix Hausdorff

A set X endowed with a system U(p) of subsets for every p 2 X is called a topological structure, provided i) p 2 U for every neighborhood U 2 U(p) ii) If U 2 U(p) and V ◆ U, then V 2 U(p)

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SLIDE 16

Topology as defined by Felix Hausdorff

A set X endowed with a system U(p) of subsets for every p 2 X is called a topological structure, provided i) p 2 U for every neighborhood U 2 U(p) ii) If U 2 U(p) and V ◆ U, then V 2 U(p) iii) If U1, U2 2 U(p), then U1 \ U2 2 U(p) and X 2 U(p)

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SLIDE 17

Topology as defined by Felix Hausdorff

A set X endowed with a system U(p) of subsets for every p 2 X is called a topological structure, provided i) p 2 U for every neighborhood U 2 U(p) ii) If U 2 U(p) and V ◆ U, then V 2 U(p) iii) If U1, U2 2 U(p), then U1 \ U2 2 U(p) and X 2 U(p) iv) For every U 2 U(p) there exists a V 2 U(p) such that U 2 U(y) for all y 2 V

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Contents

  • 1. Motivation — my early topology
  • 2. Topology

3. Interlude on prerequisites

  • 4. Cryptomorphy of topology concepts
  • 5. Continuity

6. Interlude on structure comparison 7. Interlude on the existential and inverse image

  • 8. Relating continuity with the inverse image
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Axioms

A heterogeneous relation algebra

I is a category wrt. composition “;” and identities

,

I has as morphism sets complete atomic boolean lattices

with [, \, , , , ✓,

I obeys rules for transposition T in connection with the latter

two that may be stated in either one of the following two ways: Dedekind rule: R;S \ Q ✓ (R \ Q;ST); (S \ RT;Q)

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SLIDE 20

Axioms

A heterogeneous relation algebra

I is a category wrt. composition “;” and identities

,

I has as morphism sets complete atomic boolean lattices

with [, \, , , , ✓,

I obeys rules for transposition T in connection with the latter

two that may be stated in either one of the following two ways: Dedekind rule: R;S \ Q ✓ (R \ Q;ST); (S \ RT;Q) Schr¨

  • der equivalences:

A;B ✓ C ( ) AT;C ✓ B ( ) C;BT ✓ A

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SLIDE 21

Residuals and the symmetric quotient

I R\S := RT;S left residuum

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SLIDE 22

Residuals and the symmetric quotient

I R\S := RT;S left residuum

The left residuum R\S sets into relation a column of R precisely to those columns of S containing it.

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Residuals and the symmetric quotient

I R\S := RT;S left residuum

The left residuum R\S sets into relation a column of R precisely to those columns of S containing it.

I syq(A, B) := AT;B \ A

T ;B symmetric quotient

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SLIDE 24

Residuals and the symmetric quotient

I R\S := RT;S left residuum

The left residuum R\S sets into relation a column of R precisely to those columns of S containing it.

I syq(A, B) := AT;B \ A

T ;B symmetric quotient

The symmetric quotient sets into relation equal columns.

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SLIDE 25

Illustrating the left residuum

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec US French German British Spanish B B B @ 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 C C C A A K Q J 10 9 8 7 6 5 4 3 2 US French German British Spanish B B B @ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 C C C A Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec A K Q J 10 9 8 7 6 5 4 3 2 B B B B B B B B B B B B B B B B B B B B @ 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 C C C C C C C C C C C C C C C C C C C C A

S above R below R\S Left residua show how columns of the relation R below the fraction backslash are contained in columns of the relation S above

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SLIDE 26

Illustrating the symmetric quotient

A K Q J 10 9 8 7 6 5 4 3 2 American French German British Spanish B B B @ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 C C C A Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec American French German British Spanish B B B @ 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 0 0 0 0 1 C C C A Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec A K Q J 10 9 8 7 6 5 4 3 2 B B B B B B B B B B B B B B B B B B B B @ 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 C C C C C C C C C C C C C C C C C C C C A

R above S below syq(R, S) The symmetric quotient shows which columns of the upper are equal to columns of the lower relation

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SLIDE 27

Set Comprehension

Finding equal columns i, k of relations R, S: 8n : (n, i) 2 R $ (n, k) 2 S 8n : (n, i) 2 R ! (n, k) 2 S ^ (n, i) 2 R (n, k) 2 S 8n : (n, i) 2 R ! (n, k) 2 S and 8n : (n, i) 2 R (n, k) 2 S 9n : (n, i) 2 R ^ (n, k) / 2 S and 9n (n, i) / 2 R ^ (n, k) 2 S (i, k) 2 RT;S \ R

T ;S

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SLIDE 28

Construction of domains

Given a relation algebra, we may extend it in several ways:

I direct product I direct sum I direct power I quotient I extrusion I target permutation

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Construction of domains

Given any direct products by projections π : X ⇥ Y ! X, ρ : X ⇥ Y ! Y, π0 : U ⇥ V ! U, ρ0 : U ⇥ V ! V , we define the Kronecker product, the fork-, and the join-operator: i) (A ⇥ B) := π;A;π0T \ ρ;B;ρ0T ii) (C < D) := C;πT \ D;ρT iii) (E > F) := π;E \ ρ;F

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Direct power — up to isomorphism

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Direct power — up to isomorphism

  • Any relation ε satisfying

I syq(ε, ε) ✓ ,

(i.e., in fact syq(ε, ε) = )

I syq(ε, R) is surjective for every relation R starting in X.

is called a direct power interpreted with 2-relation DirPow x P(X) Member x ε : X ! P(X)

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SLIDE 32

{} {♠} {♥} {♠,♥} {♦} {♠,♦} {♥,♦} {♠,♥,♦} {♣} {♠,♣} {♥,♣} {♠,♥,♣} {♦,♣} {♠,♦,♣} {♥,♦,♣} {♠,♥,♦,♣}

ε =

♠ ♥ ♦ ♣ B @ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 C A

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SLIDE 33

{} {♠} {♥} {♠,♥} {♦} {♠,♦} {♥,♦} {♠,♥,♦} {♣} {♠,♣} {♥,♣} {♠,♥,♣} {♦,♣} {♠,♦,♣} {♥,♦,♣} {♠,♥,♦,♣}

ε =

♠ ♥ ♦ ♣ B @ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 C A {♠,♦,♣} {♠,♥,♦} {♠,♥,♦,♣} {♠,♥,♣} {♦} {♠,♦} {♥,♦} {♠} {♥,♣} {♠,♣} {♣} {♠,♥} {♦,♣} {} {♥,♦,♣} {♥}

ε0 =

♠ ♥ ♦ ♣ B @ 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 C A

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SLIDE 34

{} {♠} {♥} {♠,♥} {♦} {♠,♦} {♥,♦} {♠,♥,♦} {♣} {♠,♣} {♥,♣} {♠,♥,♣} {♦,♣} {♠,♦,♣} {♥,♦,♣} {♠,♥,♦,♣}

ε =

♠ ♥ ♦ ♣ B @ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 C A {♠,♦,♣} {♠,♥,♦} {♠,♥,♦,♣} {♠,♥,♣} {♦} {♠,♦} {♥,♦} {♠} {♥,♣} {♠,♣} {♣} {♠,♥} {♦,♣} {} {♥,♦,♣} {♥}

ε0 =

♠ ♥ ♦ ♣ B @ 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 C A

P := syq(ε, ε0) satisfies ε;syq(ε, ε0) = ε0

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SLIDE 35

Membership relations

U = ε;e e = syq(ε, U)

{} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} a b c d    0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1   

 

1 1

 

(0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0) = eT {} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} {} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d}                         1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1                        

                   

1

                    Subset U and corresponding point e in the powerset via ε, Ω

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SLIDE 36

Contents

  • 1. Motivation — my early topology
  • 2. Topology

3. Interlude on prerequisites

  • 4. Cryptomorphy of topology concepts
  • 5. Continuity

6. Interlude on structure comparison 7. Interlude on the existential and inverse image

  • 8. Relating continuity with the inverse image
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SLIDE 37

Topology as defined by Felix Hausdorff — recalled

A set X endowed with a system U(p) of subsets for every p 2 X is called a topological structure, provided

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SLIDE 38

Topology as defined by Felix Hausdorff — recalled

A set X endowed with a system U(p) of subsets for every p 2 X is called a topological structure, provided i) p 2 U for every neighborhood U 2 U(p)

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SLIDE 39

Topology as defined by Felix Hausdorff — recalled

A set X endowed with a system U(p) of subsets for every p 2 X is called a topological structure, provided i) p 2 U for every neighborhood U 2 U(p) ii) If U 2 U(p) and V ◆ U, then V 2 U(p)

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SLIDE 40

Topology as defined by Felix Hausdorff — recalled

A set X endowed with a system U(p) of subsets for every p 2 X is called a topological structure, provided i) p 2 U for every neighborhood U 2 U(p) ii) If U 2 U(p) and V ◆ U, then V 2 U(p) iii) If U1, U2 2 U(p), then U1 \ U2 2 U(p) and X 2 U(p)

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SLIDE 41

Topology as defined by Felix Hausdorff — recalled

A set X endowed with a system U(p) of subsets for every p 2 X is called a topological structure, provided i) p 2 U for every neighborhood U 2 U(p) ii) If U 2 U(p) and V ◆ U, then V 2 U(p) iii) If U1, U2 2 U(p), then U1 \ U2 2 U(p) and X 2 U(p) iv) For every U 2 U(p) there exists a V 2 U(p) such that U 2 U(y) for all y 2 V

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SLIDE 42

Topology — lifted

i) p 2 U for every neighborhood U 2 U(p) U ✓ ε

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SLIDE 43

Topology — lifted

i) p 2 U for every neighborhood U 2 U(p) U ✓ ε ii) If U 2 U(p) and V ◆ U, then V 2 U(p) U;Ω ✓ U

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Topology — lifted

i) p 2 U for every neighborhood U 2 U(p) U ✓ ε ii) If U 2 U(p) and V ◆ U, then V 2 U(p) U;Ω ✓ U iii) If U1, U2 2 U(p), then U1 \ U2 2 U(p) and X 2 U(p) (U < U) ;M ✓ U U; =

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SLIDE 45

Topology — lifted

i) p 2 U for every neighborhood U 2 U(p) U ✓ ε ii) If U 2 U(p) and V ◆ U, then V 2 U(p) U;Ω ✓ U iii) If U1, U2 2 U(p), then U1 \ U2 2 U(p) and X 2 U(p) (U < U) ;M ✓ U U; = iv) For every U 2 U(p) there exists a V 2 U(p) such that U 2 U(y) for all y 2 V U ✓ U;εT;U

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SLIDE 46

The same with ε conceiving U as a relation:

ε : X ! 2X and U : X ! 2X “For every U 2 U(p) there exists a V 2 U(p) such that U 2 U(y) for all y 2 V ”

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SLIDE 47

The same with ε conceiving U as a relation:

ε : X ! 2X and U : X ! 2X “For every U 2 U(p) there exists a V 2 U(p) such that U 2 U(y) for all y 2 V ” 8p, U : U 2 U(p) !

  • 9V : V 2 U(p)^
  • 8y : y 2 V ! U 2 U(y)
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SLIDE 48

The same with ε conceiving U as a relation:

ε : X ! 2X and U : X ! 2X “For every U 2 U(p) there exists a V 2 U(p) such that U 2 U(y) for all y 2 V ” 8p, U : U 2 U(p) !

  • 9V : V 2 U(p)^
  • 8y : y 2 V ! U 2 U(y)
  • 8p, U : UpU !
  • 9V : UpV ^
  • 8y : εyV ! UyU
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SLIDE 49

The same with ε conceiving U as a relation:

ε : X ! 2X and U : X ! 2X “For every U 2 U(p) there exists a V 2 U(p) such that U 2 U(y) for all y 2 V ” 8p, U : U 2 U(p) !

  • 9V : V 2 U(p)^
  • 8y : y 2 V ! U 2 U(y)
  • 8p, U : UpU !
  • 9V : UpV ^
  • 8y : εyV ! UyU
  • 8p, U : UpU !
  • 9V : UpV ^ 9y : εyV ^ UyU
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SLIDE 50

The same with ε conceiving U as a relation:

ε : X ! 2X and U : X ! 2X “For every U 2 U(p) there exists a V 2 U(p) such that U 2 U(y) for all y 2 V ” 8p, U : U 2 U(p) !

  • 9V : V 2 U(p)^
  • 8y : y 2 V ! U 2 U(y)
  • 8p, U : UpU !
  • 9V : UpV ^
  • 8y : εyV ! UyU
  • 8p, U : UpU !
  • 9V : UpV ^ 9y : εyV ^ UyU
  • 8p, U : UpU !
  • 9V : UpV ^ εT;UV U
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SLIDE 51

The same with ε conceiving U as a relation:

ε : X ! 2X and U : X ! 2X “For every U 2 U(p) there exists a V 2 U(p) such that U 2 U(y) for all y 2 V ” 8p, U : U 2 U(p) !

  • 9V : V 2 U(p)^
  • 8y : y 2 V ! U 2 U(y)
  • 8p, U : UpU !
  • 9V : UpV ^
  • 8y : εyV ! UyU
  • 8p, U : UpU !
  • 9V : UpV ^ 9y : εyV ^ UyU
  • 8p, U : UpU !
  • 9V : UpV ^ εT;UV U
  • 8p, U : UpU !
  • U;εT;U
  • pU
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SLIDE 52

The same with ε conceiving U as a relation:

ε : X ! 2X and U : X ! 2X “For every U 2 U(p) there exists a V 2 U(p) such that U 2 U(y) for all y 2 V ” 8p, U : U 2 U(p) !

  • 9V : V 2 U(p)^
  • 8y : y 2 V ! U 2 U(y)
  • 8p, U : UpU !
  • 9V : UpV ^
  • 8y : εyV ! UyU
  • 8p, U : UpU !
  • 9V : UpV ^ 9y : εyV ^ UyU
  • 8p, U : UpU !
  • 9V : UpV ^ εT;UV U
  • 8p, U : UpU !
  • U;εT;U
  • pU

U ✓ U;εT;U

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SLIDE 53

A neighborhood topology and the basis of its open sets

A relation U : X ! 2X will be called a neighborhood topology if the following properties are satisfied: i) U; = and U ✓ ε, ii) U;Ω ✓ U, iii) (U < U) ;M ✓ U, iv) U ✓ U;εT;U.

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SLIDE 54

A neighborhood topology and the basis of its open sets

A relation U : X ! 2X will be called a neighborhood topology if the following properties are satisfied: i) U; = and U ✓ ε, ii) U;Ω ✓ U, iii) (U < U) ;M ✓ U, iv) U ✓ U;εT;U. U =

{} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} a b c d B @ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 C A

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SLIDE 55

Topology given by transition to the open kernel

We call a relation K : 2X ! 2X a mapping-to-open-kernel topology, if i) K is a kernel forming, i.e., K ✓ ΩT, Ω;K ✓ K;Ω, K;K ✓ K,

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SLIDE 56

Topology given by transition to the open kernel

We call a relation K : 2X ! 2X a mapping-to-open-kernel topology, if i) K is a kernel forming, i.e., K ✓ ΩT, Ω;K ✓ K;Ω, K;K ✓ K, contracting isotonic idempotent ii) ε;KT is total, iii) (K ⇥ K);M ✓ M;K;ΩT, in fact (K ⇥ K);M = M;K.

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SLIDE 57

Topology given by transition to the open kernel

We call a relation K : 2X ! 2X a mapping-to-open-kernel topology, if i) K is a kernel forming, i.e., K ✓ ΩT, Ω;K ✓ K;Ω, K;K ✓ K, contracting isotonic idempotent ii) ε;KT is total, iii) (K ⇥ K);M ✓ M;K;ΩT, in fact (K ⇥ K);M = M;K. kernel forming commutes with intersection

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SLIDE 58

A topology in different forms

  • {}

{a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} a b c d    0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1    {} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} a b c d    0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1    {} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} a b c d    0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1    {} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} {} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d}                         1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1                                                  1 1 1 1 1 1 1 1 1 1                         

ε U εO := ε ∩

;OT

V = ε;K ∩ ε

K := syq(U, ε) indicating OD as diagonal OV

slide-59
SLIDE 59

Non-topological kernel-forming

K =

{} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} {} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d}                         1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1                        

Kernel-forming that is not a topology, since not intersection-closed

slide-60
SLIDE 60

{} {1} {2} {1,2} {3} {1,3} {2,3} {1,2,3} {4} {1,4} {2,4} {1,2,4} {3,4} {1,3,4} {2,3,4} {1,2,3,4} 1 2 3 4 B @ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 C A

slide-61
SLIDE 61

{} {1} {2} {1,2} {3} {1,3} {2,3} {1,2,3} {4} {1,4} {2,4} {1,2,4} {3,4} {1,3,4} {2,3,4} {1,2,3,4} {5} {1,5} {2,5} {1,2,5} {3,5} {1,3,5} {2,3,5} {1,2,3,5} {4,5} {1,4,5} {2,4,5} {1,2,4,5} {3,4,5} {1,3,4,5} {2,3,4,5} {1,2,3,4,5} 1 2 3 4 5 B B B @ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 C C C A

slide-62
SLIDE 62

{} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} a b c d B @ 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 C A

slide-63
SLIDE 63

Cryptomorphy of diverse topology concepts

U 7! K := syq(U, ε) : 2X ! 2X K 7! U := ε;KT : X ! 2X. OD 7! U := ε;OD;Ω K, U 7! OD := \ εT;U = KT;K

slide-64
SLIDE 64

Cryptomorphy of diverse topology concepts

U 7! K := syq(U, ε) : 2X ! 2X K 7! U := ε;KT : X ! 2X. OD 7! U := ε;OD;Ω K, U 7! OD := \ εT;U = KT;K This means the obligation to prove, e.g. U; = , U ✓ ε, U;Ω ✓ U, (U < U) ;M ✓ U, U ✓ U;εT;U. ( ) K ✓ ΩT, Ω;K ✓ K;Ω, K;K ✓ K, ε;KT; = , (K ⇥ K);M = M;K.

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SLIDE 65

Separation axioms

Let a topology on X be given via neighborhoods, open sets, kernel mapping as required. It is T0-space (sometimes a Kolmogorov space) if for any two points in X an open set exists that contains one of them but not the other. It is T1-space when 8x, y : x = / y ! 9U, V 2 O : x 2 U ^ y 2 / U ^ y 2 V ^ x 2 / V . It is T2-space, i.e., a topology satisfying the Hausdorff property, when 8x, y : x = / y ! 9U, V 2 O : x 2 U ^ y 2 V ^ ; = U \ V .

slide-66
SLIDE 66

Separation axioms

Let a topology given in relational form, i.e., by U, O, K, εO as

  • required. It is called a

i) T0-space if syq(U T, U T) = ii) T1-space if ✓ U;U

T.

iii) T2-space or a Hausdorff space if ✓ U;εT;ε;U T.

slide-67
SLIDE 67

Contents

  • 1. Motivation — my early topology
  • 2. Topology

3. Interlude on prerequisites

  • 4. Cryptomorphy of topology concepts
  • 5. Continuity

6. Interlude on structure comparison 7. Interlude on the existential and inverse image

  • 8. Relating continuity with the inverse image
slide-68
SLIDE 68

Continuity — standard vs. relational definition

  • T
  • Let any two neighborhood topologies U, U0 be given on sets

X, X0, and a mapping f : X ! X0. f continuous :( ) For p 2 X and every neighborhood U0 2 U0(f(p)), there exists a neighborhood U 2 U(p) satisfying f(U) ✓ U0.

slide-69
SLIDE 69

f =

a b c d 1 2 3 4 5 B B B @ 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 C C C A

  • ϑf =

{} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} {} {1} {2} {1,2} {3} {1,3} {2,3} {1,2,3} {4} {1,4} {2,4} {1,2,4} {3,4} {1,3,4} {2,3,4} {1,2,3,4} {5} {1,5} {2,5} {1,2,5} {3,5} {1,3,5} {2,3,5} {1,2,3,5} {4,5} {1,4,5} {2,4,5} {1,2,4,5} {3,4,5} {1,3,4,5} {2,3,4,5} {1,2,3,4,5}                                                       1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1                                                      

slide-70
SLIDE 70

f =

a b c d e 1 2 3 4 5 B B B @ 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 C C C A

slide-71
SLIDE 71

Contents

  • 1. Motivation — my early topology
  • 2. Topology

3. Interlude on prerequisites

  • 4. Cryptomorphy of topology concepts
  • 5. Continuity

6. Interlude on structure comparison 7. Interlude on the existential and inverse image

  • 8. Relating continuity with the inverse image
slide-72
SLIDE 72

Structure-preserving mappings

Let be given two “structures” of whatever kind abstracted to relations R1 : X1 ! Y1 and R2 : X2 ! Y2.

1

R

1

Y

1

X

2

X

2

Y

2

R

slide-73
SLIDE 73

Structure-preserving mappings

Let be given two “structures” of whatever kind abstracted to relations R1 : X1 ! Y1 and R2 : X2 ! Y2.

Φ Ψ

1

  • 1
  • 1
  • 2
  • 2
  • 2
  • Given mappings Φ : X1

! X2 and Ψ : Y1 ! Y2, we may ask whether these mappings transfer the first structure “sufficiently nice” into the second one.

slide-74
SLIDE 74

Structure-preserving mappings

Let be given two “structures” of whatever kind abstracted to relations R1 : X1 ! Y1 and R2 : X2 ! Y2.

Φ Ψ

1

  • 1
  • 1
  • 2
  • 2
  • 2
  • Given mappings Φ : X1

! X2 and Ψ : Y1 ! Y2, we may ask whether these mappings transfer the first structure “sufficiently nice” into the second one. If any two elements x, y are in relation R1, then their images Φ(x), Ψ(y) shall be in relation R2.

slide-75
SLIDE 75

Structure-preserving mappings

Let be given two “structures” of whatever kind abstracted to relations R1 : X1 ! Y1 and R2 : X2 ! Y2.

Φ Ψ

1

  • 1
  • 1
  • 2
  • 2
  • 2
  • Given mappings Φ : X1

! X2 and Ψ : Y1 ! Y2, we may ask whether these mappings transfer the first structure “sufficiently nice” into the second one. If any two elements x, y are in relation R1, then their images Φ(x), Ψ(y) shall be in relation R2. 8x 2 X1 : 8y 2 Y1 : (x, y) 2 R1 ! (Φ(x), Ψ(y)) 2 R2

slide-76
SLIDE 76

Structure-preserving mappings

Let be given two “structures” of whatever kind abstracted to relations R1 : X1 ! Y1 and R2 : X2 ! Y2.

Φ Ψ

1

  • 1
  • 1
  • 2
  • 2
  • 2
  • Given mappings Φ : X1

! X2 and Ψ : Y1 ! Y2, we may ask whether these mappings transfer the first structure “sufficiently nice” into the second one. If any two elements x, y are in relation R1, then their images Φ(x), Ψ(y) shall be in relation R2. 8x 2 X1 : 8y 2 Y1 : (x, y) 2 R1 ! (Φ(x), Ψ(y)) 2 R2 R1;Ψ ✓ Φ;R2

slide-77
SLIDE 77

Homomorphism

This concept works for groups, fields and other algebraic structures, but also for relational structures as, e.g., graphs. Φ, Ψ is a homomorphism from R to R0, if Φ, Ψ are mappings satisfying R;Φ ✓ Ψ;R0. Φ, Ψ is an isomorphism between R and R0, if Φ, Ψ as well as ΦT, ΨT are homomorphisms.

Theorem

If Φ, Ψ are mappings, then R;Ψ ✓ Φ;R0 ( ) R ✓ Φ;R0;ΨT ( ) ΦT;R ✓ R0;ΨT ( ) ΦT;R;Ψ ✓ R0 If relations Φ, Ψ are not mappings, one cannot fully execute this rolling; there remain different forms of (bi-)simulations.

slide-78
SLIDE 78

Continuity compares structures in a different way!

  • T
  • Let any two neighborhood topologies U, U0 be given on sets

X, X0, and a mapping f : X ! X0. f continuous :( ) For p 2X and every neighborhood U0 2 U0(f(p)), there exists a neighborhood U 2 U(p) satisfying f(U) ✓ U0.

slide-79
SLIDE 79

Contents

  • 1. Motivation — my early topology
  • 2. Topology

3. Interlude on prerequisites

  • 4. Cryptomorphy of topology concepts
  • 5. Continuity

6. Interlude on structure comparison 7. Interlude on the existential and inverse image

  • 8. Relating continuity with the inverse image
slide-80
SLIDE 80

Existential image of relations

  • T
slide-81
SLIDE 81

Existential image of relations

  • T
  • ϑ := ϑR := syq(RT;ε, ε0)

existential image.

slide-82
SLIDE 82

Existential image of relations

  • T
  • ϑ := ϑR := syq(RT;ε, ε0)

existential image. ϑ is (lattice-)continuous wrt. the powerset orderings Ω = εT;ε

slide-83
SLIDE 83

Existential image of relations

  • T
  • ϑ := ϑR := syq(RT;ε, ε0)

existential image. ϑ is (lattice-)continuous wrt. the powerset orderings Ω = εT;ε ϑ

X =

2X

ϑQ;R = ϑQ

;ϑR

i.e. multiplicative εT;R = ϑR

;ε0T

ε0T;RT = ϑRT ;εT i.e. mutual simulation R may be re-obtained from ϑ as R = ε;ϑ;ε0T

slide-84
SLIDE 84

Existential image of relations

  • T
  • ϑ := ϑR := syq(RT;ε, ε0)

existential image. ϑ is (lattice-)continuous wrt. the powerset orderings Ω = εT;ε ϑ

X =

2X

ϑQ;R = ϑQ

;ϑR

i.e. multiplicative εT;R = ϑR

;ε0T

ε0T;RT = ϑRT ;εT i.e. mutual simulation R may be re-obtained from ϑ as R = ε;ϑ;ε0T but there exist many relations W satisfying R = ε;W ;ε0T

slide-85
SLIDE 85

Existential image

  • T
  • R =

a b c d 1 2 3 4 5     0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0    

ϑR =

{} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d} {} {1} {2} {1,2} {3} {1,3} {2,3} {1,2,3} {4} {1,4} {2,4} {1,2,4} {3,4} {1,3,4} {2,3,4} {1,2,3,4} {5} {1,5} {2,5} {1,2,5} {3,5} {1,3,5} {2,3,5} {1,2,3,5} {4,5} {1,4,5} {2,4,5} {1,2,4,5} {3,4,5} {1,3,4,5} {2,3,4,5} {1,2,3,4,5}                                                       1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1                                                      

slide-86
SLIDE 86

Inverse image

R =

a b c d 1 2 3 4 5     0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0    

  • T
  • {

}  

ϑRT =

{} {1} {2} {1,2} {3} {1,3} {2,3} {1,2,3} {4} {1,4} {2,4} {1,2,4} {3,4} {1,3,4} {2,3,4} {1,2,3,4} {5} {1,5} {2,5} {1,2,5} {3,5} {1,3,5} {2,3,5} {1,2,3,5} {4,5} {1,4,5} {2,4,5} {1,2,4,5} {3,4,5} {1,3,4,5} {2,3,4,5} {1,2,3,4,5} {} {a} {b} {a,b} {c} {a,c} {b,c} {a,b,c} {d} {a,d} {b,d} {a,b,d} {c,d} {a,c,d} {b,c,d} {a,b,c,d}                         1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1                        

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SLIDE 87

Contents

  • 1. Motivation — my early topology
  • 2. Topology

3. Interlude on prerequisites

  • 4. Cryptomorphy of topology concepts
  • 5. Continuity

6. Interlude on structure comparison 7. Interlude on the existential and inverse image

  • 8. Relating continuity with the inverse image
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SLIDE 88

Continuity compares structures in a different way!

  • T
  • Let any two neighborhood topologies U, U0 be given on sets

X, X0, and a mapping f : X ! X0. f continuous :( ) For p 2X and every neighborhood U0 2 U0(f(p)), there exists a neighborhood U 2 U(p) satisfying f(U) ✓ U0.

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SLIDE 89

Lifting the continuity condition

For all p 2X, all V 2 U0(f(p)), exists a U 2 U(p) with f(U) ✓ V . 8p 2 X : 8V 2 U0(f(p)) : 9U 2 U(p) : f(U) ✓ V 8p 2 X : 8v 2 2X0 : U0

f(p),v

!

  • 9u : Up,u ^

⇥ 8y : εyu ! ε0

f(y),v

⇤ 8p : 8v : (f;U0)pv !

  • 9u : Upu ^

⇥ 8y : εyu ! (f;ε0)yv ⇤ 8p : 8v : (f;U0)pv !

  • 9u : Upu ^ 9y : εyu ^ (f;ε0)yv
  • 8p : 8v : (f;U0)pv

!

  • 9u : Upu ^ εT;f;ε0uv
  • 8p : 8v : (f;U0)pv

!

  • U;εT;f;ε0

pv

f;U0 ✓ U;εT;f;ε0 f;U0 ✓ U;ϑT

fT

The last step is proved as follows: U;εT;f;ε0 ✓ U;εT;f;ε0;ϑfT;ϑT

fT

because ϑfT is total = U;εT;f;ε0;syq(f;ε0, ε);ϑT

fT

by definition of ϑfT ✓ U;εT;ε;ϑT

fT

cancellation = U;εT;ε;ϑT

fT

since ϑfT is a mapping = U;Ω;ϑT

fT = U;ϑT fT

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SLIDE 90

Continuity — standard vs. relational definition

  • T
  • Let any two neighborhood topologies U, U0 be given on sets

X, X0, and a mapping f : X ! X0. f continuous :( ) For p 2 X and every neighborhood U0 2 U0(f(p)), there exists a neighborhood U 2 U(p) satisfying f(U) ✓ U0. f continuous :( ) f;U0;ϑfT ✓ U ( ) f;U0 ✓ U;ϑT

fT

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SLIDE 91

Cryptomorphy of continuity concepts

Given sets X, X0 with topologies, we consider a mapping f : X ! X0 together with its inverse image ϑfT : 2X0 ! 2X. Then we say that the pair (f, ϑfT) is i) K-continuous :( ) KT

2;ϑfT ✓ ε2

T;f T;ε1;KT

1

ii) OD-continuous :( ) OD2;ϑfT ✓ ϑfT;OD1 iii) OV -continuous :( ) ϑT

fT;O0 V ✓ OV

iv) εO-continuous :( ) f;εO2;ϑfT ✓ εO1

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SLIDE 92

Cryptomorphy of continuity concepts

Given sets X, X0 with topologies, we consider a mapping f : X ! X0 together with its inverse image ϑfT : 2X0 ! 2X. Then we say that the pair (f, ϑfT) is i) K-continuous :( ) KT

2;ϑfT ✓ ε2

T;f T;ε1;KT

1

ii) OD-continuous :( ) OD2;ϑfT ✓ ϑfT;OD1 iii) OV -continuous :( ) ϑT

fT;O0 V ✓ OV

iv) εO-continuous :( ) f;εO2;ϑfT ✓ εO1 Again, there is an obligation to prove f is K-continuous ( ) f is OD-continuous ( ) f is OV -continuous ( ) f is εO-continuous

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SLIDE 93

Thank you!

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SLIDE 94

Language and system

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SLIDE 95

Systems to support work with relations

I RelView: RBDD-Implementierung; auch f¨

ur große Relationen

I TITUREL eine relationale Sprache, transformierbar,

interpretierbar

I Ralf: weiland ein guter Formel-Manipulator und

Beweis-Assistent

I RATH: Exploring (finite) relation algebras with tools

written in Haskell

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SLIDE 96

Aims in designing TITUREL

I Formulate all problems so far tackled with relational

methods

I Transform relational terms and formulae in order to

  • ptimize them

I Interpret the relational constructs as boolean matrices, in

RelView, in the TITUREL substrate, or in Rath

I Prove relational formulae with system support in the style

  • f Ralf or Rasiowa-Sikorski

I Translate relational formulae into T

EX-representation, or to first-order predicate logic, e.g.

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SLIDE 97

Recalling syntax vs. semantics for PL/I:

tokens K const. ϕ fct. p pred. interpretation I in supporting set an element KI for K a function table ϕI for ϕ s subset pI for p

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SLIDE 98

Recalling syntax vs. semantics for PL/I:

tokens K const. ϕ fct. p pred. interpretation I in supporting set an element KI for K a function table ϕI for ϕ s subset pI for p Out of this and the variables V one forms terms and formulae T = V | K | ϕ(T) F = p(T) | ¬F | 8V : F

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SLIDE 99

Recalling syntax vs. semantics for PL/I:

tokens K const. ϕ fct. p pred. interpretation I in supporting set an element KI for K a function table ϕI for ϕ s subset pI for p Out of this and the variables V one forms terms and formulae T = V | K | ϕ(T) F = p(T) | ¬F | 8V : F With a variable valuation v : x 7! v(x) terms are evaluated v⇤(x) := v(x) v⇤(k) := kI v⇤(ϕ(t)) := ϕI(v⇤(t))

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SLIDE 100

Recalling syntax vs. semantics for PL/I:

tokens K const. ϕ fct. p pred. interpretation I in supporting set an element KI for K a function table ϕI for ϕ s subset pI for p Out of this and the variables V one forms terms and formulae T = V | K | ϕ(T) F = p(T) | ¬F | 8V : F With a variable valuation v : x 7! v(x) terms are evaluated v⇤(x) := v(x) v⇤(k) := kI v⇤(ϕ(t)) := ϕI(v⇤(t)) and formulae interpreted | =I,v p(t) :( ) v⇤(t) ✓ pI | =I,v ¬F :( ) | = / I,vF | =I,v 8x : F :( ) For all s holds | =I,vx s F

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SLIDE 101

Relational language

ground

finite baseset with named/numbered elements

ground

catO elem bs and predicate/marking/ listing/powerset element etc.

ground

vect bs + bs + matrix/set function/ predicate/pairlist etc.

ground

rela sets of rela, vect, elem formulae

DirSum/Prod DirPow QuotMod

bs with named/numbered element

InjFrom

Language Language Substrate Substrate

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SLIDE 102

The system TITUREL runs under one of the following acronym interpretations

  • This is the ultimate relation system
  • Towards improved techniques using relations
  • Teaching informaticians to use relations
  • Try it, to use relations
  • Toolkit intended to use relations
  • Testing innovative tools using relations
  • Think innovative - try using relations
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SLIDE 103

TITUREL ontvangt de Heilige

Graal en de Heilige Speer uit handen van een Engelenschaar die neder daalt uit de hemel. Hij bouwt een Tempel voor deze heilige relikwien, de Graalburcht Montsalvat. Ridders die tot de Graal worden geroepen vormen de ridderschap van de Heilige Graal, hun Koning is Titurel. Op hoge leeftijd draagt hij zijn ambt

  • ver op zijn zoon Amfortas.
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SLIDE 104

Model questions

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SLIDE 105

Model problem

Theory of relation algebra Matrix- algebras of relations

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SLIDE 106

Model problem

Theory of relation algebra Matrix algebras of relations

McKenzie algebra

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SLIDE 107

Predicate logic vs. relational logic

RRA (representable relation algebras, i.e. the Boolean matrix algebras) are not finitely axiomatizable. (Don Monk) RA can express any (and up to logical equivalence, exactly the) first-order logic formulas containing no more than three variables. RRA is axiomatizable by a universal Horn theory.

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SLIDE 108

Model problem

a c b ab b c a bc bc

v

a c ab

v

a c

T =

aT = c bT = b cT = a a2 = a c2 = c b2 = b = [ a [ c a;c = c;a = a;b = b;a = a [ b c;b = b;c = c [ b Ralph McKenzie’s homogeneous non-representable RA The element a cannot be conceived as a Boolean matrix.

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SLIDE 109

Model problem

Matrix- algebras of relations Theory of relation algebra

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SLIDE 110

Model problem

Matrix- algebras of relations Theory of relation algebra

# (R;P ⇥ S;Q) ✓ (R ⇥ S); (P ⇥ Q)

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SLIDE 111

Model problem

Matrix- algebras of relations Theory of relation algebra

# (R;P ⇥ S;Q) ✓ (R ⇥ S); (P ⇥ Q)

  • "

(R;P ⇥ S;Q) ◆ (R ⇥ S); (P ⇥ Q)

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SLIDE 112

Model problem

Matrix- algebras of relations Theory of relation algebra

π;R;P ;π00T \ ρ;S;Q;ρ00T✓ # (π;R;π0T \ ρ;S;ρ0T);(π0;P ;π00T \ ρ0;Q;ρ00T)

  • "

π;R;P ;π00T \ ρ;S;Q;ρ00T◆ (π;R;π0T \ ρ;S;ρ0T);(π0;P ;π00T \ ρ0;Q;ρ00T)

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SLIDE 113

Model problem

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SLIDE 114

Model problem

It is, however, possible to prove that (Q ⇥

X); ( B

⇥ R) = (Q ⇥ R) = ( A ⇥ R); (Q ⇥

Y )

This does express correctly that Q and R may with one execution thread be executed in either order; i.e., with meandering “coroutines”. But no two execution threads are provided to execute in parallel.

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SLIDE 115

Model problem

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SLIDE 116

Model problem

Q P S R

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SLIDE 117

Model problem

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SLIDE 118

History of relations

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SLIDE 119

History of relations

Relations were being developed at a time when

I formal semantics was not yet known

language and interpretation typing and unification

I the idea that several models of a theory may exist, was

close to being completely unknown (non-Euclidian geometry: Bolyai, Lobatschevskij ⇡ 1840)

I one was still bound to handle the following in the

respective natural language, namely in English, German, Latin, Greek, Japanese, Russian, Arabic . . . ! quantification 8, 9 conversion RT composition A ;B but also ”brother“, ”father“, ”uncle“ and only gradually developed a more standardized language

I the concept of a matrix had not yet been coined

(Cayley, Sylvester 1850’s)

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SLIDE 120

History of relations

George Boole’s investigations on the laws of thought of 1854: In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects

  • f its operation are confined. The most unfettered discourse is

that in which the words we use are understood in the widest possible application, and for them the limits of discourse are co-extensive with those of the universe itself. But more usually we confine ourselves to a less spacious field. . . . Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse. The office of any name or descriptive term employed under the limitations supposed is not to raise in the mind the conception of all the beings or objects to which that name or description is applicable, but only of those which exist within the supposed universe of discourse.

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SLIDE 121

History of relations

  • quantifjcation

typing matrices theorem K

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SLIDE 122

Closure and contact

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SLIDE 123

Closure and contact

Definition

Let some ordered set (V, ) be given. A mapping ρ : V ! V is called a closure operation, if it is i) expanding x  ρ(x), ii) isotonic x  y ! ρ(x)  ρ(y), iii) idempotent ρ(ρ(x))  ρ(x).

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SLIDE 124

Closure and contact

Definition

Let some ordered set (V, ) be given. A mapping ρ : V ! V is called a closure operation, if it is i) expanding x  ρ(x), ii) isotonic x  y ! ρ(x)  ρ(y), iii) idempotent ρ(ρ(x))  ρ(x). As usual: quantifiers omitted. We now reinstall them

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SLIDE 125

Closure and contact

Definition

Let some ordered set (V, ) be given. A mapping ρ : V ! V is called a closure operation, if it is i) expanding x  ρ(x), ii) isotonic x  y ! ρ(x)  ρ(y), iii) idempotent ρ(ρ(x))  ρ(x). As usual: quantifiers omitted. We now reinstall them 8x, y : x  y ! ρ(x)  ρ(y)

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SLIDE 126

Closure and contact

Definition

Let some ordered set (V, ) be given. A mapping ρ : V ! V is called a closure operation, if it is i) expanding x  ρ(x), ii) isotonic x  y ! ρ(x)  ρ(y), iii) idempotent ρ(ρ(x))  ρ(x). As usual: quantifiers omitted. We now reinstall them 8x, y : x  y ! ρ(x)  ρ(y) which makes 18 symbols in standard mathematics notation. This will now shrink down to just 7.

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SLIDE 127

Theorem

Assume an ordering E : X ! X and a mapping ρ : X ! X. Then ρ is a closure operator if and only if ρ ✓ E E;ρ ✓ ρ;E ρ;ρ ✓ ρ

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SLIDE 128

Theorem

Assume an ordering E : X ! X and a mapping ρ : X ! X. Then ρ is a closure operator if and only if ρ ✓ E E;ρ ✓ ρ;E ρ;ρ ✓ ρ We convince ourselves, that the intentions of the preceding definition are met when lifting in this way, starting from ρ(ρ(x))  ρ(x): 8x, y, z : ρxy ^ ρyz ! [9w : ρxw ^ Ezw] ( ) 8x, y, z : ρxy ^ ρyz ! (ρ;ET)xz ( ) ¬

  • 9x, z : (9y : ρxy ^ ρyz) ^ [ρ;ET]xz
  • (

) ¬

  • 9x, z : (ρ;ρ)xz ^ [ρ;ET]xz
  • (

) 8x, z : (ρ;ρ)xz ! [ρ;ET]xz ( ) ρ;ρ ✓ ρ;ET

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SLIDE 129

Theorem

Assume an ordering E : X ! X and a mapping ρ : X ! X. Then ρ is a closure operator if and only if ρ ✓ E E;ρ ✓ ρ;E ρ;ρ ✓ ρ We convince ourselves, that the intentions of the preceding definition are met when lifting in this way, starting from ρ(ρ(x))  ρ(x): 8x, y, z : ρxy ^ ρyz ! [9w : ρxw ^ Ezw] ( ) 8x, y, z : ρxy ^ ρyz ! (ρ;ET)xz ( ) ¬

  • 9x, z : (9y : ρxy ^ ρyz) ^ [ρ;ET]xz
  • (

) ¬

  • 9x, z : (ρ;ρ)xz ^ [ρ;ET]xz
  • (

) 8x, z : (ρ;ρ)xz ! [ρ;ET]xz ( ) ρ;ρ ✓ ρ;ET Together with the others, we get ( ) ρ;ρ ✓ ρ

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SLIDE 130

Closure and contact

Definition

We consider a set related to its powerset, with a membership relation ε : X ! P(X) and a powerset ordering Ω : P(X) ! P(X). A relation C : X ! P(X) is called an Aumann contact relation, provided i) it contains the membership relation, i.e., ε ✓ C, ii) an element x in contact with a set Y all of whose elements are in contact with a set Z, will be in contact with Z, the so-called infectivity of contact, i.e., C;εT;C ✓ C, or equivalently, CT;C ✓ εT;C. One will easily show that C forms an upper cone, i.e., C;Ω ✓ C: CT;C ✓ εT;C ✓ εT;ε = Ω

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SLIDE 131

Closure and contact

Theorem

Given a closure operator ρ : P(X) ! P(X) on some powerset defined via a membership relation ε : X ! P(X), the construct C := ε;ρT turns out to be an Aumann contact relation.

Beweis.

i) ε ✓ ε;ρT ( ) ε;ρ ✓ ε ( = ε;Ω ✓ ε. ii) C;εT;C = ε;ρT;εT;ε;ρT = ε;ρT;εT;ε;ρT since ρ is a mapping = ε;ρT;Ω;ρT ✓ ε;Ω;ρT;ρT with the second closure property ✓ ε;Ω;ρT with the third closure property = ε;ρT = C since ε;Ω = ε

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SLIDE 132

Closure and contact

Theorem

Given any Aumann contact relation C : X ! P(X), forming the construct ρ := syq(C, ε) results in a closure operator. Proof: i) ρ = syq(C, ε) ✓ CT;ε ✓ εT;ε = Ω ii) We recall ε;syq(ε, Y ) = Y and ε;syq(ε, Y ) = Y for ρ;Ω;ρT = syq(C, ε);εT;ε;syq(ε, C) = CT;C ✓ εT;ε = Ω. Since ρ is a mapping, we may proceed with ρ;Ω;ρT ✓ Ω Ω ✓ ρ;Ω;ρT Ω;ρ ✓ ρ;Ω iii) We prove ρ;ρ ✓ ρ, i.e., syq(C, ε);syq(C, ε) ✓ syq(C, ε) or (C

T ;ε [ CT;ε);syq(ε, C) ✓ C T ;ε [ CT;ε

Now, the two terms on the left are treated separately.

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SLIDE 133

Example

Let an arbitrary relation R : X ! Y be given.

  • 2

Then C := R;R

T ;ε is always an Aumann contact relation. To

show this, we have to prove ε ✓ R;R

T ;ε = C, which is trivial using Schr¨

  • der equivalences.

CT;C ✓ εT;C ( ) R;R

T ;ε T ;R;R T ;ε ✓ εT;R;R T ;ε

( = R;R

T ;ε T ;R ✓ εT;R

( ) εT;R;R

T ✓ (R;R T ;ε)T

The construct C := R;R

T ;ε may be read as follows: It declares

those combinations x 2 X and S ✓ X to be in contact C, for which every relationship (x, y) 2 / R implies that there exists also an x0 2 S in relation (x0, y) 2 / R.

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SLIDE 134

Exzerpt of bibliography of trade union publication