Distance Theory Algebraically Supplemented
1
Distance Preparatory consideration
1
Language is regarded as set X.
Product set X×X
Map d from product set X×X to R+ = { x∈R ; x≧0 } satisfies next 3conditions ( 3 axiom of distance).
(1) d (x, y) ≧ 0, d (x, y) = 0 ⇔ x = y
(2) d (x, y) = d (y, x) <Symmetry>
(3) d (x, y) ≦ d (x, y) + d (y, z) <Triangle inequality>
d is called <distance function> or <metric function>.
Set ( X, d ) is called< metric space>.
2
Set X
Point of X a
Plus real number r
Set { x∈X ; d ( a, x ) < r } is called <open ball> with radius r centered by a. Expression is D ( a, r )
Word is regarded as open ball D.
3
Metric space ( X, d )
Subset of X A
Arbitrary point of A a
Open ball centered by a D⊂A
A is called <open set>.
All of As U
U satisfies next 3 conditions ( 3 axiom of open set ).
(1) 0, X ∈U
(2) U1, …, Uk ∈U ⇒⋂k i=1 Uk ∈U
(3) Uα∈U , α∈Λ ⇒ ∪α∈Λ ∈U
U is called <topology> or <open set system> over X.
Set ( X, U ) is called <topological space>.
Sentence is regarded as topological space.
4
Topological space ( X, U ) defined by metric space ( X, d ) is <first axiom of countability>.
5
Topological space ( X, U ) that is defined by metric space ( X, d ) is <Hausdorff space>.
[Note] Hausdorff separation axiom “ Toward two points x≠y, two neighborhoods Ux and Uy have existence never crossing each other.”
Tokyo October 8, 2007
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