Distance Theory Algebraically Supplemented 1 Distance Preparatory consideration

Distance Theory Algebraically Supplemented

1

Distance Preparatory consideration

TANAKA Akio

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.

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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, …, U_k ∈U ⇒⋂^k _i=1 U_k ∈U

(3) U_α∈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 U_x and U_y have existence never crossing each other.”

Tokyo October 8, 2007

Sekinan Research Field of Language