Wednesday 24 July 2013

Distance Theory Algebraically Supplemented 3 Point Preparatory consideration

Point Preparatory consideration

Continuous map     X  Y
Inverse continuous map      f -1 : Y  X
f ( or f -1 ) is homeomorphism.
X and Y are homeomorphic.
Two topological space     XY
Continuous map    fi X    ( = 0,1)
Family of continuous maps      ft X  Y    t[0,1] )
Existence of maps is homotopic.
Expression is f0  f1
ft[ 0,1] ) is homotopy.
<Homotopy equivalent>
Two topological space     XY
Continuous map    X  Y  g X  Y
Composition     X  X   Y  Y
Homotopy     ≅ Id X   ≅ IdY
<Topological pair>
Topological space     X
Topological subspace     A
Topological pair     ( X)
<Attaching space>
Two topological space     XY
XY = Ø
Homeomorphism (and attaching map) BA
Attaching space    XhY
<Cell complex>
n-dimensional disk     Dn   (n1)
n-dimensional sphere     Sn-1  (n1)
Dn = Sn-1
i-dimensional disk     Di
Inside of Di     Di ∂Di Di – Sn-1
Homeomorphism at inside of Di    cell    (0i1)  Expression is ei.
Closed cell of ei     ēi    
ēi ēi ei
Topological space     X
Sum of cells     X= ē01ē0m
Attaching map     h1 : X(1)  X0    
Attaching space     X1:= X0h1 X(1)
XXn-1hn X(n)
X = Xn
X is n-dimensional cell complex (or cell complex).
Topological space     X 
Subspace A that has only a point     x0     xis called base point.)
Topological space pair     ( Xx)
Homotopy set     [ (InIn), ( Xx) ]     n; natural number
The set is group.
n-dimensional sphere     Sn
A base point on Sn     x0
Topological space pair     (Sn, x0)
Isomorphism      [ (InIn), ( Xx) ]  [ (Sn, x0), ( Xx) ]
Arbitrary two points in X     x0x1
Natural number     n
Isomorphism      [ (InIn), ( Xx) ]  [ (InIn), ( Xx1) ]
(InIn), ( Xx) ] that is entered group structure is called 1-dimensional homotopy group     π1 ( Xx)
π1 ( Xx) is called fundamental group.
π1 ( Xx { 1 } is called simply connected.
n-dimensional homotopy group is commutative group.
Two topological space pairs ( Xx), ( Yy) are homotopy equivalent.
Isomorphism    πn ( Xxπn ( Yy)
Arbitrary two points in X     x0x1
Isomorphism     h : ( Xx ( Xx)
Homotopy equivalent
Isomorphism πn( Xx ( Xx)
<Convex set>
Vector space over real number field     X
Arbitrary points in K     xy
K is convex set when what x and y make line segment is contained in K.
(n+1) points in space Rn     p0, …, pn
Minimal vertex set     σn = ( p0, …, p) is called n-simplex. n is dimension ofσn.
Face ofσn     j-simplexσj = ( p01, …, pnj )
Boundary ofσn     all the simplexes less or equal (n-1) dimensions     Expression is ∂σn.
<Simplicial complex>
Simplex in Rn     σn
Set of simplexes     S = {σn }
S satisfied by next conditions is Simplicial complex.
(1) σnS  all the faces ofσnS
(2) σm1,σn2S →σm1⋂σn2 is face ofσm1 and σn2.
Topological space that consists of sum of all the simplexes
Topological space     X
Simplicial complex     K
Polyhedron of K     | K |
Making K that is homeomorphic between | K | and X is triangulation.
Topological space is regarded as simplicial complex.
<Simplicial complex homology group>
Simplicial complexes makes simplicial complex homology group by definition of equivalent relation.
From Simplicial complex, composed simplexes are divided to smaller simplexes.
<Simplicial map>
By subdivision, continuous map from simplicial complexes to another one can be approximated by simplicial approximation theorem.
<Simplicial approximation theorem>
Topological space     X
Simplicial complex     K
Polyhedron of K    | K |
Homeomorphic map     : | K |  X
Triangulation     T = ( K)
Topological space     X1X2
Triangulation      T 1= ( K1t1 )     T 2= ( K2t2 )
Simplicial map     f : X1  X2
Simplicial map on T1T2     φX1  X2
Point     xX1
φ (x)simplex of T2
φ is smplicial approximation of f.
Existence of φ is called simplicial approximation theorem.
<Dimension axiom of homology group>
Topological space pair      ( XA )
Commutative group     hXA )  (p = 0,1,2,…)
Topological space being consisted of a point pt  when p1   hp (pt) = 0
<CW complex>
Cell complex     X
When X satisfies next two conditions, is CW complex.
(1) Closed cell ē of X’s cell is contained in sum-set of finite cells. The closed cell is called <closure finite>.
(2) Subset of X    U
Toward cell of X, when U  ē is open set of eU is only open set at the time. This situation is called <weak topology>.
CW complex is locally contractible in paracompact normal space.
<Locally contractible>
Paracompact is that locally finite is given by adequate <refinement> toward arbitrary open covering in topological space.
Normal space is satisfies T2 (Fréchet separation axiom) and T( Tietze separation axiom) in topological space.
Topological space     X
A point of X     p
Arbitrary neighborhood of p     U
Open neighborhood of p     V
Toward U, there exists that inclusion map i :  and Constant map care homotopic.
From Hausdorff space ( topological T2 space ) to complex ( especially CW complex ), bridge is algebraically built by approach between space and a point.

Tokyo October 12, 2007

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