compliments to bad spaces - rasolegviro/compl2badspaces.pdf · differential geometry. cambridge...

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Compliments to Bad Spaces

Oleg Viro

March 13, 2007

Human factor

• Human factor

Differential Spaces

Finite TopologicalSpaces

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The ways that mathematical theories find to the core ofmainstream mathematical curriculums are strongly influencedby accident circumstances.

Human factor

• Human factor

Differential Spaces


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The ways that mathematical theories find to the core ofmainstream mathematical curriculums are strongly influencedby accident circumstances.Individuals shape Mathematics.

Human factor

• Human factor

Differential Spaces


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The ways that mathematical theories find to the core ofmainstream mathematical curriculums are strongly influencedby accident circumstances.Individuals shape Mathematics.The shapes are not perfect.

Human factor

• Human factor

Differential Spaces


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The ways that mathematical theories find to the core ofmainstream mathematical curriculums are strongly influencedby accident circumstances.Individuals shape Mathematics.The shapes are not perfect.Often basic definitions could be made more convenient thanthe present ones.

Human factor

• Human factor

Differential Spaces


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I do not mean to criticize Calculus, although it is awful.

Human factor

• Human factor

Differential Spaces


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Take fresher examples:

Human factor

• Human factor

Differential Spaces


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Take fresher examples:differentiable manifolds and finite topological spaces.

Human factor

• Human factor

Differential Spaces


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I am going to emphasize opportunities

Human factor

• Human factor

Differential Spaces


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I am going to emphasize opportunities,but need to motivate the positive things by some criticism.

Human factor

• Human factor

Differential Spaces


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I am going to emphasize opportunities,but need to motivate the positive things by some criticism.

The opportunities are not lost yet.

Differential Spaces

• Human factor

Differential Spaces

• DifferentiableManifolds

• What is wrong

• Political correctnessin Mathematics

• Differential Structures

• Differential Spaces

• Differentiable maps

• Generating, refining,relaxing

• Subspaces

• Embeddings

• Example

• Constructing newdifferential spaces

• Examples of quotientspaces• Tangent vectors anddimensions

• Metric spaces


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Differentiable Manifolds

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The modern definition of differential manifold was given in thebook by O. Veblen and J.H.C. Whitehead The foundations ofdifferential geometry. Cambridge tracts in mathematics andmathermatical physics.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The modern definition of differential manifold was given in thebook by O. Veblen and J.H.C. Whitehead The foundations ofdifferential geometry. Cambridge tracts in mathematics andmathermatical physics.Published in 1932 by Cambridge University Press.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The modern definition of differential manifold was given in thebook by O. Veblen and J.H.C. Whitehead The foundations ofdifferential geometry. Cambridge tracts in mathematics andmathermatical physics.Published in 1932 by Cambridge University Press.Inspired by H.Weyl’s book on Riemann surfaces Die Idee derRiemannschen Flache published in 1913.

What is wrong

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The traditional definition of smooth structures is quite long

What is wrong

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The traditional definition of smooth structures is quite longand different from definitions of similar and closely relatedstructures

What is wrong

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The traditional definition of smooth structures is quite longand different from definitions of similar and closely relatedstructures studied in algebraic geometry and topology.

What is wrong

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Smooth structures are traditionally defined only on manifolds.

What is wrong

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Smooth structures are traditionally defined only on manifolds.This deprives us of flexibility that we enjoy in general topology

What is wrong

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Smooth structures are traditionally defined only on manifolds.This deprives us of flexibility that we enjoy in general topology,where any set-theoretic construction has a topologicalcounter-part:

What is wrong

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Smooth structures are traditionally defined only on manifolds.This deprives us of flexibility that we enjoy in general topology,where any set-theoretic construction has a topologicalcounter-part:a subset 7→ a subspace,

What is wrong

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Smooth structures are traditionally defined only on manifolds.This deprives us of flexibility that we enjoy in general topology,where any set-theoretic construction has a topologicalcounter-part:a subset 7→ a subspace,a quotient set 7→ a quotient space, etc.

What is wrong

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The image of a differential manifold under a differentiable mapmay be not a manifold,

What is wrong

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The image of a differential manifold under a differentiable mapmay be not a manifold, and hence not eligible to bear any traceof a differential structure.

Political correctness in Mathematics

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Terminology related to differentiable manifolds does not let usspeak on bad spaces.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Terminology related to differentiable manifolds does not let usspeak on bad spaces.Is this good?


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Terminology related to differentiable manifolds does not let usspeak on bad spaces.Is this good?Is this acceptable?


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Terminology related to differentiable manifolds does not let usspeak on bad spaces.Is this good?Is this acceptable?Even if you hate pathology


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Terminology related to differentiable manifolds does not let usspeak on bad spaces.Is this good?Is this acceptable?Even if you hate pathology , do you know beforehandwhat is pathologically bad in Mathematics?


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Terminology related to differentiable manifolds does not let usspeak on bad spaces.Is this good?Is this acceptable?Even if you hate pathology , do you know beforehandwhat is pathologically bad in Mathematics?Would you like to have an ability to speak aboutthe natural smooth structure on CP 2/ conj ?


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Terminology related to differentiable manifolds does not let usspeak on bad spaces.Is this good?Is this acceptable?Even if you hate pathology , do you know beforehandwhat is pathologically bad in Mathematics?Would you like to have an ability to speak aboutthe natural smooth structure on CP 2/ conj ?smooth structure on a Cantor set?


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Terminology related to differentiable manifolds does not let usspeak on bad spaces.Is this good?Is this acceptable?Even if you hate pathology , do you know beforehandwhat is pathologically bad in Mathematics?Would you like to have an ability to speak aboutthe natural smooth structure on CP 2/ conj ?smooth structure on a Cantor set? , a fractal set?


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The notion of differential space was developed in the sixties


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The notion of differential space was developed in the sixties,but has not found a way to the mainstream Mathematics.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Why?


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Why? Was it not a right time for this?


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Why? Was it not a right time for this?Were there not right people?

Differential Structures

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X be a set and r be a natural number or ∞ .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X be a set and r be a natural number or ∞ .A differential structure of class Cr on X


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X be a set and r be a natural number or ∞ .A differential structure of class Cr on X

not differentiable, but differential,for nobody is going to differentiate it!


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X be a set and r be a natural number or ∞ .A differential structure of class Cr on X is an algebraCr(X) of functions X → R such that:


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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1.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

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• Example



• Metric spaces


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1. Composition of functions belonging to Cr(X) withCr -differentiable function belongs to Cr(X) .


• Human factor

Differential Spaces


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In other words, (g ◦ f : X → R) ∈ Cr(X)


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In other words, (g ◦ f : X → R) ∈ Cr(X)if f : X → U is defined by f1, . . . , fn ∈ Cr(X) ,U ⊂ R

n is an open set,


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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n is an open set,and g : U → R is a Cr -map.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



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2. f ∈ Cr(X) if near each point of X it coincides with afunction belonging to Cr(X) .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

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2. f ∈ Cr(X) if near each point of X it coincides with afunction belonging to Cr(X) .

In other words, f ∈ Cr(X) if for each a ∈ X there existg, h ∈ Cr(X) such that h(a) > 0 and f(x) = g(x)for each x with h(x) > 0 .

Differential Spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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A pair consisting of a set X and a differential structure ofclass Cr on X is called a differential space of class Cr

Differential Spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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A pair consisting of a set X and a differential structure ofclass Cr on X is called a differential space of class Cr , orjust a Cr-space.

Differential Spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Examples

Differential Spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Examples1. Any smooth manifold X with algebra Cr(X) ofCr-differentiable functions.

Differential Spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Examples1. Any smooth manifold X with algebra Cr(X) ofCr-differentiable functions.2. Discrete space. Any X and all functions X → R .

Differential Spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Examples1. Any smooth manifold X with algebra Cr(X) ofCr-differentiable functions.2. Discrete space. Any X and all functions X → R .3. Indiscrete space. Any X and all constant functionsX → R .

Differential Spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Examples1. Any smooth manifold X with algebra Cr(X) ofCr-differentiable functions.2. Discrete space. Any X and all functions X → R .3. Indiscrete space. Any X and all constant functionsX → R .4. Topological space. A topological space X with allcontinuous functions X → R .

Differentiable maps

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X and Y be Cr-spaces.

Differentiable maps

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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f : X → Y is called a Cr-mapif f ◦ φ ∈ Cr(X) for any φ ∈ Cr(Y ) .

Differentiable maps

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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A Cr-map f : X → Y induces f∗ : Cr(Y ) → Cr(X) .

Differentiable maps

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Cr-spaces and Cr-maps constitute a category.

Differentiable maps

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Cr-spaces and Cr-maps constitute a category.

Isomorphisms of the category are called Cr-diffeomorphims.

Generating, refining, relaxing

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For any set F of real valued functions on a set X ,there exists a minimal Cr-structure on X containing F .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For any set F of real valued functions on a set X ,there exists a minimal Cr-structure on X containing F .It is said to be generated by F .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For example, coordinate projections Rn → R generate thestandard differential structure on Rn .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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The Cr-structure generated by a Cs-structure C with s < rcoincides with C .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For example, a C0-structure


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For example, a C0-structure

which is nothing but a topological structure.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For example, a C0-structure is a Cr-structure for any r .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For example, a C0-structure is a Cr-structure for any r .On the other hand, when decreasing r ,we have to add new functions.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



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A Cr-structure A generated as a Cr-structure by aCs-structure B with s > r is called a relaxation of B .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

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A Cr-structure A generated as a Cr-structure by aCs-structure B with s > r is called a relaxation of B .Then B is called a refinement of A .

Subspaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X be a differential space and A ⊂ X .

Subspaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X be a differential space and A ⊂ X .Restrictions to A of functions differentiable on X

Subspaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X be a differential space and A ⊂ X .Restrictions to A of functions differentiable on Xdo not necessarily constitute a differential structure on A .

Subspaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For example, if X = R and A = R>0 = {x | x > 0} ,

Subspaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For example, if X = R and A = R>0 = {x | x > 0} ,the function A → R : x 7→ 1

xis not a restriction of any

function continuous on R ,

Subspaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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function continuous on R ,but in a neighborhood of any point it is a restriction of afunction differentiable on R .

Subspaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Restrictions to A of functions differentiable on X generate adifferential structure.

Subspaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Restrictions to A of functions differentiable on X generate adifferential structure. This structure is said to be induced on Aby the structure of X

Subspaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Restrictions to A of functions differentiable on X generate adifferential structure. This structure is said to be induced on Aby the structure of X ,and A equipped with this structure is called a (differential)subspace of X.

Subspaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Restrictions to A of functions differentiable on X generate adifferential structure. This structure is said to be induced on Aby the structure of X ,and A equipped with this structure is called a (differential)subspace of X.

Whitney Problem: Describe the differential structure inducedon a closed X ⊂ R

n .

Embeddings

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X and Y be differential spaces (of class Cr).

Embeddings

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X and Y be differential spaces (of class Cr).A map f : X → Y is called a differential embeddingif it defines a diffeomorphism X → f(X) .

Embeddings

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Let X and Y be differential spaces (of class Cr).A map f : X → Y is called a differential embeddingif it defines a diffeomorphism X → f(X) .(Here f(X) is considered as a differential subspace of Y ).

Embeddings

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For a differential space X , functions f1, . . . , fn define adifferential embeddingf : X → R

n : x 7→ (f1(x), . . . , fn(x))

Embeddings

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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For a differential space X , functions f1, . . . , fn define adifferential embeddingf : X → R

n : x 7→ (f1(x), . . . , fn(x))iff f1, . . . , fn generate Cr(X) and f is injective.

Example

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


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Consider the set C of all differentiable functions R → R withderivative vanishing at 0 .

Example

• Human factor

Differential Spaces


• What is wrong






• Subspaces

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• Metric spaces


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Consider the set C of all differentiable functions R → R withderivative vanishing at 0 . This is a differential structure.

Example

• Human factor

Differential Spaces


• What is wrong






• Subspaces

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• Metric spaces


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Consider the set C of all differentiable functions R → R withderivative vanishing at 0 . This is a differential structure.How does the space (R, C) look like?Is it embeddable to R

2 ?

Example

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Differential Spaces


• What is wrong






• Subspaces

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2 ?

We need functions u, v : R → R with u′(0) = v′(0) = 0such that any differential function f : R → R with f ′(0) = 0was a composition F ◦ (u × v) for some differentiableF : R

2 → R .

Example

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Differential Spaces


• What is wrong






• Subspaces

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2 ?


2 → R .

Take u(x) = x2 , v(x) = x3 .

Example

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Differential Spaces


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2 ?


2 → R .

Take u(x) = x2 , v(x) = x3 .

A parametrization of semicubical parabola:

Constructing new differential spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

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• Metric spaces


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Multiplication. Let X and Y be Cr-spaces.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

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• Metric spaces


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Multiplication. Let X and Y be Cr-spaces.The canonical way to define Cr-structure in X × Y


• Human factor

Differential Spaces


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Multiplication. Let X and Y be Cr-spaces.The canonical way to define Cr-structure in X × Yis to generate it by{f ◦ prX | f ∈ Cr(X)} ∪ {g ◦ prY | g ∈ Cr(Y )} .


• Human factor

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Factorization. Let X be a Cr-space and

∼ be an equivalence relation in X .


• Human factor

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∼ be an equivalence relation in X .The Cr-structure in the quotient set X/∼canonically defined by Cr(X)


• Human factor

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∼ be an equivalence relation in X .The Cr-structure in the quotient set X/∼canonically defined by Cr(X)consists of f : X/∼ → R such that(f ◦ pr : X → R) ∈ Cr(X) .

Examples of quotient spaces

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Differential Spaces


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1. What differential space is obtained by identification of theend points of [0, 1] ?


• Human factor

Differential Spaces


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1. What differential space is obtained by identification of theend points of [0, 1] ? Is it embeddable to R2 ?


• Human factor

Differential Spaces


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If so, how does the the image look like?


• Human factor

Differential Spaces


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If so, how does the the image look like? Like this: ?


• Human factor

Differential Spaces


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• Subspaces

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No, like this: !


• Human factor

Differential Spaces


• What is wrong






• Subspaces

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No, like this: ! Or that: !


• Human factor

Differential Spaces


• What is wrong






• Subspaces

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No, like this: ! Or that: ! But not this: !


• Human factor

Differential Spaces


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2. What if we take [0, 1.5] and identifyeach x ∈ [0, 0.5] with x + 1 ?


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Differential Spaces


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Then we get really a space diffeomorphic to .


• Human factor

Differential Spaces


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New factorization:


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Differential Spaces


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New factorization:Identifying end points of [0, 1] , identify also tangent vectors!


• Human factor

Differential Spaces


• What is wrong






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• Example



• Metric spaces


15 / 25






New factorization:Identifying end points of [0, 1] , identify also tangent vectors!That is consider functions f : [0, 1] → R with

f(0) = f(1) and f ′(0) = f ′(1) .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


15 / 25







f(0) = f(1) and f ′(0) = f ′(1) .

The resulting space: .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


15 / 25







f(0) = f(1) and f ′(0) = f ′(1) .

The resulting space: . Smooth, but with jump ofcurvature.

Tangent vectors and dimensions

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25

It is easier to define cotangent vectors.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25

It is easier to define cotangent vectors.Let X be a differential space and p ∈ X .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25

It is easier to define cotangent vectors.Let X be a differential space and p ∈ X . Functionsvanishing at p form a maximal ideal mp of R-algebraCr(X) .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25

It is easier to define cotangent vectors.Let X be a differential space and p ∈ X . Functionsvanishing at p form a maximal ideal mp of R-algebraCr(X) . The cotangent space T ∗

p (X) is mp/m2

p .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .

Tangent space Tp(X) is the dual to T ∗

p (X) .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .


p (X) .It can be defined as the space of differentiations ofdifferentiable functions on X at p .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .


p (X) .It can be defined as the space of differentiations ofdifferentiable functions on X at p . As usual.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .



Other traditional definition of tangent vectors (via anequivalence of smooth paths)


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .



Other traditional definition of tangent vectors (via anequivalence of smooth paths) gives another result


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .



Other traditional definition of tangent vectors (via anequivalence of smooth paths) gives another result and doesnot give a vector space.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .



dim T ∗

p (X) may differ from the topological dimensionof X at p .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .



dim T ∗

p (X) may differ from the topological dimensionof X at p . For example, dim T0([0, 1]/(0 ∼ 1)) = 2 .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .



dim T ∗


Theorem: If Cr(X) is the set of all continuous functions on atopological space X , then dim T ∗

p (X) = 0.


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .



dim T ∗



p (X) = 0.

The quotient space D2/∂D2 of disk D2 is homeomorphicto sphere S2 .


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .



dim T ∗



p (X) = 0.

The quotient space D2/∂D2 of disk D2 is homeomorphicto sphere S2 . What is dim∂D2/∂D2(D2/∂D2) ?


• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


16 / 25


p (X) is mp/m2

p .



dim T ∗



p (X) = 0.

The quotient space D2/∂D2 of disk D2 is homeomorphicto sphere S2 . What is dim∂D2/∂D2(D2/∂D2) ? Infinity!

Metric spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


17 / 25

Each metric space is a differential space.

Metric spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


17 / 25

Each metric space is a differential space.A metric gives rise to many functions:

Metric spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


17 / 25

Each metric space is a differential space.A metric gives rise to many functions: distances from points.

Metric spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


17 / 25

Each metric space is a differential space.A metric gives rise to many functions: distances from points.However on a Riemannian manifold they are not differentiable.

Metric spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


17 / 25


In a sufficiently small neighborhood of a point, distances fromother points form local coordinate system.

Metric spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


17 / 25



Let X be a metric space. A function f : X → R isdifferentiable at p ∈ X if for any neighborhood U of p thereexist points q1, . . . , qn ∈ U and real numbers a1 , . . . , an

such that

|f(x) − f(p) −∑

ai(dist(qi, x) − dist(qi, p))|

dist(x, p)→ 0

as x → p .

Metric spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


17 / 25




such that

|f(x) − f(p) −∑


dist(x, p)→ 0

as x → p . Is this definition good?

Metric spaces

• Human factor

Differential Spaces


• What is wrong






• Subspaces

• Embeddings

• Example



• Metric spaces


17 / 25




such that

|f(x) − f(p) −∑


dist(x, p)→ 0

as x → p . Is this definition good? At least,it recovers the smooth structure of a Riemannian manifold.

Finite Topological Spaces

• Human factor

Differential Spaces


• Hesitation of finitespaces

• Fundamental group

• Space of faces

• Homotopy

• Digital plane andJordan Theorem

• Arbitrary finite space

• Baricentricsubdivision

18 / 25

Hesitation of finite spaces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25

Topology seams to be the only fields in Mathematics thathesitates of its own finite objects, finite topological spaces.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25

Topology seams to be the only fields in Mathematics thathesitates of its own finite objects, finite topological spaces.Finite sets, finite dimensional vector spaces, finite fields, finiteprojective spaces, etc. are appreciated by their host theories.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25


Who is guilty?


• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25


Who is guilty? Interest towards Analysis?


• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25


Who is guilty? Interest towards Analysis? Hausdorff axiom?


• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25


Who is guilty? Interest towards Analysis? Hausdorff axiom?Topology textbooks?


• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25



An average mathematician is well aware at bestabout two kinds of finite topological spaces:


• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25



An average mathematician is well aware at bestabout two kinds of finite topological spaces:discrete and indiscrete.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25




Let us take a look at the rest of them.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25




Let us take a look at the rest of them.They are not that bad!


• Human factor

Differential Spaces




• Space of faces

• Homotopy




19 / 25




Let us take a look at the rest of them.They are not that bad!At early days of topology, they were the main objects of theCombinatorial Topology.

Fundamental group

• Human factor

Differential Spaces




• Space of faces

• Homotopy




20 / 25

What is the minimal number of points in a topological spacewith nontrivial fundamental group?

Fundamental group

• Human factor

Differential Spaces




• Space of faces

• Homotopy




20 / 25


What is the group?

Fundamental group

• Human factor

Differential Spaces




• Space of faces

• Homotopy




20 / 25


What is the group?

What is the next group?

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25

Let P be a compact polyhedron.

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25

Let P be a compact polyhedronrepresented as the union of closed convex polyhedraany two of which meet in a common face.

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25


P is partitioned to open faces of these convex polyhedrons.

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25


P is partitioned to open faces of these convex polyhedrons.The quotient space Q is a finite topological space.

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P .

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P .Especially if the partition was a triangulation.

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P . Each point of Q represents aface of P .

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P . Each point of Q represents aface of P . Points representing vertices are closed.

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P . Each point of Q represents aface of P . Points representing vertices are closed.The closure of a point x ∈ Q consists of points correspondingto the faces of the corresponding face of P .

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P . Each point of Q represents aface of P . Points representing vertices are closed.The closure of a point x ∈ Q consists of points correspondingto the faces of the corresponding face of P .Each point in a finite space has minimal neighborhood.

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P . Each point of Q represents aface of P . Points representing vertices are closed.The closure of a point x ∈ Q consists of points correspondingto the faces of the corresponding face of P .Each point in a finite space has minimal neighborhood,the intersection of all of its neighborhoods.

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P . Each point of Q represents aface of P . Points representing vertices are closed.The closure of a point x ∈ Q consists of points correspondingto the faces of the corresponding face of P .Each point in a finite space has minimal neighborhood.In Q the minimal neighborhood of a point corresponds to thestar of corresponding face.

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P . Each point of Q represents aface of P . Points representing vertices are closed.The closure of a point x ∈ Q consists of points correspondingto the faces of the corresponding face of P .Each point in a finite space has minimal neighborhood.In Q the minimal neighborhood of a point corresponds to thestar of corresponding face.The star St(σ) of a face σ is the union of all faces Σsuch that ∂Σ ⊃ σ .

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P . Each point of Q represents aface of P . Points representing vertices are closed.The closure of a point x ∈ Q consists of points correspondingto the faces of the corresponding face of P .Each point in a finite space has minimal neighborhood.In Q the minimal neighborhood of a point corresponds to thestar of corresponding face. Faces in P are partially orderedby adjacency: Σ > σ iff Cl(Σ) ⊃ σ .

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P . Each point of Q represents aface of P . Points representing vertices are closed.The closure of a point x ∈ Q consists of points correspondingto the faces of the corresponding face of P .Each point in a finite space has minimal neighborhood.In Q the minimal neighborhood of a point corresponds to thestar of corresponding face. Faces in P are partially orderedby adjacency: Σ > σ iff Cl(Σ) ⊃ σ .This partial order defines and is defined by the topology of Q .

Space of faces

• Human factor

Differential Spaces




• Space of faces

• Homotopy




21 / 25



Q knows everything on P . Each point of Q represents aface of P . Points representing vertices are closed.The closure of a point x ∈ Q consists of points correspondingto the faces of the corresponding face of P .Each point in a finite space has minimal neighborhood.In Q the minimal neighborhood of a point corresponds to thestar of corresponding face. Faces in P are partially orderedby adjacency: Σ > σ iff Cl(Σ) ⊃ σ .This partial order defines and is defined by the topology of Q .

P can be recovered from Q .

Homotopy

• Human factor

Differential Spaces




• Space of faces

• Homotopy




22 / 25

Let P be a triangulated polyhedron

Homotopy

• Human factor

Differential Spaces




• Space of faces

• Homotopy




22 / 25

Let P be a triangulated polyhedron,

Q the space of its simplices

Homotopy

• Human factor

Differential Spaces




• Space of faces

• Homotopy




22 / 25


Q the space of its simplices (the quotient space of P )

Homotopy

• Human factor

Differential Spaces




• Space of faces

• Homotopy




22 / 25


Q the space of its simplices (the quotient space of P ),

pr : P → Q the natural projection.

Homotopy

• Human factor

Differential Spaces




• Space of faces

• Homotopy




22 / 25




For topological spaces X and Y denote by π(X,Y )the set of homotopy classes of maps X → Y .

Homotopy

• Human factor

Differential Spaces




• Space of faces

• Homotopy




22 / 25





Theorem. For any topological space X , composition with prdefines a bijection π(X,P ) → π(X,Q) .

Homotopy

• Human factor

Differential Spaces




• Space of faces

• Homotopy




22 / 25






Corollary. All homotopy and singular homology groups of Pand Q are isomorphic.

Homotopy

• Human factor

Differential Spaces




• Space of faces

• Homotopy




22 / 25






Corollary. All homotopy and singular homology groups of Pand Q are isomorphic.

Corollary. Any compact polyhedron is weak homotopyequivalent to a finite topological space.

Digital plane and Jordan Theorem

• Human factor

Differential Spaces




• Space of faces

• Homotopy




23 / 25


• Human factor

Differential Spaces




• Space of faces

• Homotopy




23 / 25

Digital line D is the quotient space of R by partition to pointsof Z and open intervals (n, n + 1) .


• Human factor

Differential Spaces




• Space of faces

• Homotopy




23 / 25


Digital plane is D2 .


• Human factor

Differential Spaces




• Space of faces

• Homotopy




23 / 25


Digital plane is D2 . It is the quotient space of R2 by the

partition formed by integer points, open unit intervalsconnecting them, and open unit squares.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




23 / 25




Digital circle of length d is the quotient space of the circleS1 ⊂ C by the partition formed by complex roots of unity ofdegree d and open arcs connecting the roots next to eachother.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




23 / 25




Digital circle


• Human factor

Differential Spaces




• Space of faces

• Homotopy




23 / 25




Digital Jordan Theorem. (Khalimsky, Kiselman) A digitalcircle embedded in the digital plane divides it into twoconnected sets.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




23 / 25




Digital Jordan Theorem. (Khalimsky, Kiselman) A digitalcircle embedded in the digital plane divides it into twoconnected sets.

Not finite, but locally finite.

Arbitrary finite space

• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25

In any topological space there is T0-equivalence relation:

x ∼ y if x and y have the same neighborhoods.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25


x ∼ y if x and y have the same neighborhoods.The quotient space by the T0-equivalence relation satisfiesthe Kolmogorov separation axiom T0:


• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25


x ∼ y if x and y have the same neighborhoods.The quotient space by the T0-equivalence relation satisfiesthe Kolmogorov separation axiom T0:for any pair of points x , y at least one of themhas a neighborhood not containing the other one.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25


x ∼ y if x and y have the same neighborhoods.The quotient space by the T0-equivalence relation satisfiesthe Kolmogorov separation axiom T0.In any T0-space the relation x ∈ Cl y is a partial order.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25


x ∼ y if x and y have the same neighborhoods.The quotient space by the T0-equivalence relation satisfiesthe Kolmogorov separation axiom T0.In any T0-space the relation x ∈ Cl y is a partial order.Remark. Without T0 axiom this is only a preorder, that istransitive and reflexive, but not antisymmetric.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25


x ∼ y if x and y have the same neighborhoods.The quotient space by the T0-equivalence relation satisfiesthe Kolmogorov separation axiom T0.In any T0-space the relation x ∈ Cl y is a partial order.Remark. Without T0 axiom this is only a preorder, that istransitive and reflexive, but not antisymmetric(if x and y are T0-equivalent, then both x ∈ Cl y andy ∈ Cl x ).


• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25



Any partial order defines a poset topology generated by sets{x | a ≺ x} .


• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25




A topology is a poset topology iff the Kolmogorov axiom holdstrue and each point has the smallest neighborhood.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25





In particular, topology in a finite space is a poset topology iffthis is a T0-space.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25





In particular, topology in a finite space is a poset topology iffthis is a T0-space.An arbitrary finite topological space is composed of clusters ofT0-equivalent points.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




24 / 25





In particular, topology in a finite space is a poset topology iffthis is a T0-space.An arbitrary finite topological space is composed of clusters ofT0-equivalent points. The clusters are partially ordered andthe order determines the topology.

Baricentric subdivision

• Human factor

Differential Spaces




• Space of faces

• Homotopy




25 / 25

How far is a poset topology from the face space of apolyhedron?


• Human factor

Differential Spaces




• Space of faces

• Homotopy




25 / 25

How far is a poset topology from the face space of apolyhedron? Not really far, just one step construction.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




25 / 25

How far is a poset topology from the face space of apolyhedron? Let (X,≺) be a poset.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




25 / 25

How far is a poset topology from the face space of apolyhedron? Let (X,≺) be a poset. ConsiderX ′ = {a1 ≺ a2 ≺ · · · ≺ an | ai ∈ X} .


• Human factor

Differential Spaces




• Space of faces

• Homotopy




25 / 25

How far is a poset topology from the face space of apolyhedron? Let (X,≺) be a poset. ConsiderX ′ = {a1 ≺ a2 ≺ · · · ≺ an | ai ∈ X} , the set of allnon-empty finite subsets of X in each of which ≺ defines alinear order.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




25 / 25

How far is a poset topology from the face space of apolyhedron? Let (X,≺) be a poset. ConsiderX ′ = {a1 ≺ a2 ≺ · · · ≺ an | ai ∈ X} , the set of allnon-empty finite subsets of X in each of which ≺ defines alinear order. X ′ is partially ordered by inclusion.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




25 / 25

How far is a poset topology from the face space of apolyhedron? Let (X,≺) be a poset. ConsiderX ′ = {a1 ≺ a2 ≺ · · · ≺ an | ai ∈ X} , the set of allnon-empty finite subsets of X in each of which ≺ defines alinear order. X ′ is partially ordered by inclusion.Poset (X ′,⊂) is called the baricentric subdivision of (X,≺) .


• Human factor

Differential Spaces




• Space of faces

• Homotopy




25 / 25

How far is a poset topology from the face space of apolyhedron? Let (X,≺) be a poset. ConsiderX ′ = {a1 ≺ a2 ≺ · · · ≺ an | ai ∈ X} , the set of allnon-empty finite subsets of X in each of which ≺ defines alinear order. X ′ is partially ordered by inclusion.Poset (X ′,⊂) is called the baricentric subdivision of (X,≺) .The baricentric subdivision of any finite poset is the space ofsimplices of a compact triangulated polyhedron.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




25 / 25

How far is a poset topology from the face space of apolyhedron? Let (X,≺) be a poset. ConsiderX ′ = {a1 ≺ a2 ≺ · · · ≺ an | ai ∈ X} , the set of allnon-empty finite subsets of X in each of which ≺ defines alinear order. X ′ is partially ordered by inclusion.Poset (X ′,⊂) is called the baricentric subdivision of (X,≺) .The baricentric subdivision of any finite poset is the space ofsimplices of a compact triangulated polyhedron.This construction is used in combinatorics to define homologygroups of a poset.


• Human factor

Differential Spaces




• Space of faces

• Homotopy




25 / 25

How far is a poset topology from the face space of apolyhedron? Let (X,≺) be a poset. ConsiderX ′ = {a1 ≺ a2 ≺ · · · ≺ an | ai ∈ X} , the set of allnon-empty finite subsets of X in each of which ≺ defines alinear order. X ′ is partially ordered by inclusion.Poset (X ′,⊂) is called the baricentric subdivision of (X,≺) .The baricentric subdivision of any finite poset is the space ofsimplices of a compact triangulated polyhedron.This construction is used in combinatorics to define homologygroups of a poset.

Theorem. Any finite topological space is weak homotopyequivalent to a compact polyhedron.

compliments to bad spaces - rasolegviro/compl2badspaces.pdf · differential geometry. cambridge...

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