Distance Theory Algebraically Supplemented
2
Space Preparatory consideration
1
From <separation axiom>, topological space X is differentiated.
T1 <Frechét separation axiom> Set consisted from one point {x}⊂X is closed set.
T2 <Hausdorff space> x≠y x,y∈X x∈U, y∈V, U⋂V=Ø Open sets U,V exist.
T3 <Regular space> Closed sets F, Open sets U, V x⊂U, F⊂V, U⋂V = Ø
T4 <Normal space> Closed sets F,G Open sets U, V F⊂U, G⊂V, U⋂V = Ø
[Note]
Separation axiom is hypothetical conditions by which topological space can separate points or subsets from open set.
2
Set X
Family of subsets of X {Mλ}λ∈Λ
Sum-set of {Mλ}λ∈Λ ∪λ∈ΛMλ
When ∪λ∈ΛMλ is equal to X, family of subsets of X i.e.{Mλ}λ∈Λ is called <covering>.
When all the elements of family is open subsets, covering is called <open covering>.
3
Set X
Arbitrary open covering of X U={Uα ; α∈Λ}
Against finite α1, …, αk∈Λ, X⊂∪ki=1 Uαi .This is abstraction of <Heine-Borel’s theorem>.
X is called <compact space>.
4
In <axiom of choice>, compact subsets of Hausdorff space is closed sets.
[Note]
Axiom of choice is next.
Set A≠Ø
Elements of A a≠Ø
Map f : A →Sum-set∪A
Toward all the elements x of A, f (x) ∈x exists.
5
In <axiom of choice>, compact Hausdorff space is normal space T4.
In T4, Closed sets F,G Open sets U, V F⊂U, G⊂V, U⋂V = Ø.
Compact Hausdorff space is regarded as <language>.
Open sets U and V are regarded as two different <words>.
6
Product space X = Πi∈I Xi in family of compact spaces < Xi ; i∈I>
In <axiom of choice>, product space X is compact. This is <Tikchonov’s theorem>.
Product space X is regarded as <sentence>.
Tokyo October 9, 2007
Sekinan Research Field of Language
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