Wednesday 24 July 2013

Distance Theory Algebraically Supplemented 1 Distance Preparatory consideration

Distance Preparatory consideration

Language is regarded as set X.
Product set     X×X
Map d from product set X×X to R+ = { xx0 } satisfies next 3conditions ( 3 axiom of distance).
(1) d (xy 0, d (xy) = 0  x = y
(2) d (xy) = d (yx)   <Symmetry> 
(3) d (xy d (xy) + d (yz)   <Triangle inequality>
d is called <distance function> or <metric function>.
Set ( X) is called< metric space>.
Set     X
Point of X     a
Plus real number    r
Set { xa) < r } is called <open ball> with radius r centered by a.     Expression is D ( a, r )
Word is regarded as open ball D.
Metric space     ( Xd )
Subset of X     A
Arbitrary point of     a
Open ball centered by a     DA
A is called <open set>.
All of As     U
satisfies next 3 conditions ( 3 axiom of open set ).
(1) 0, X U
(2) U1, …, Uk ⇒⋂k i=1 UU
(3) UαU , αΛ  αΛ U
U is called <topology> or <open set system> over X.
Set ( XU ) is called <topological space>.
Sentence is regarded as topological space.
Topological space ( X) defined by metric space ( Xd ) is <first axiom of countability>.
Topological space ( X) that is defined by metric space ( Xd ) is <Hausdorff space>.
[Note] Hausdorff separation axiom “ Toward two points xy, two neighborhoods Ux and Uy have existence never crossing each other.”

Tokyo October 8, 2007

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