Wednesday 24 July 2013

Distance Theory Algebraically Supplemented 2 Space Preparatory consideration 9th For KARCEVSKIJ Sergej

Space Preparatory consideration

From <separation axiom>, topological space X is differentiated.
T1 <Frechét separation axiom> Set consisted from one point {x}X is closed set.
T<Hausdorff spacexy  x,yX  xUyVUV=Ø   Open sets U,exist.
T3 <Regular space> Closed sets F,  Open sets U, V    xU,  FV, U= Ø
T4 <Normal space> Closed sets F,G  Open sets U, V    FU,  GV, U= Ø
Separation axiom is hypothetical conditions by which topological space can separate points or subsets from open set.
Set     X
Family of subsets of X     {Mλ}λΛ
Sum-set of {Mλ}λΛ      λΛMλ
When λΛMλ is equal to X, family of subsets of X i.e.{Mλ}λΛ is called <covering>.
When all the elements of family is open subsets, covering is called <open covering>.
Set     X
Arbitrary open covering of X     U={Uα αΛ}
Against finite α1, …, αkΛ, Xki=1 Uαi .This is abstraction of <Heine-Borel’s theorem>.
X is called <compact space>.
In <axiom of choice>, compact subsets of Hausdorff space is closed sets.
Axiom of choice is next.
Set     A≠Ø
Elements of A    a≠Ø
Map f A Sum-setA
Toward all the elements of A, f (xx exists.
In <axiom of choice>, compact Hausdorff space is normal space T4.
In T4, Closed sets F,G  Open sets U, V    FU,  GV,  U= Ø.
Compact Hausdorff space is regarded as <language>.
Open sets U and V are regarded as two different <words>.
Product space X = ΠiI X in family of compact spaces < Xi ; iI>
In <axiom of choice>, product space is compact. This is <Tikchonov’s theorem>.
Product space X is regarded as <sentence>.

Tokyo October 9, 2007

No comments:

Post a Comment