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Injective Spaces. Part II 1

and [16] provide the notation and terminology for this paper. The following propositions are true: 1. INJECTIVE SPACES (1) For every point p of the Sierpiński space such that p = 0 holds {p} is closed. (2) For every point p of the Sierpiński space such t

JOURNALOFFORMALIZEDMATHEMATICS

Volume11,Released1999,Published2003

Inst.ofComputerScience,Univ.ofBia ystok

InjectiveSpaces.PartII1ArturKorni owiczUniversityofBia ystokJaros awGrykoUniversityofBia ystokMMLIdenti er:WAYBEL25.WWW:http://doc.docsou.com/JFM/Vol11/waybel25.htmlThearticles[26],[11],[33],[34],[35],[8],[10],[9],[7],[28],[1],[21],[22],[24],[18],[25],[23],

[14],[13],[27],[37],[15],[32],[2],[3],[4],[36],[12],[19],[29],[5],[30],[20],[31],[6],[17],and[16]providethenotationandterminologyforthispaper.

1.

Thefollowingpropositionsaretrue:

(1)

(2)ForeverypointpoftheSierpi´nskispacesuchthatp=0holds{p}isclosed.ForeverypointpoftheSierpi´nskispacesuchthatp=1holds{p}isnonclosed.INJECTIVESPACES

LetusobservethattheSierpi´nskispaceisnonT1.

Letusnotethateverytop-latticewhichiscompleteandScottisalsodiscernible.

LetusmentionthatthereexistsaT0-spacewhichisinjectiveandstrict.

Letusnotethatthereexistsatop-latticewhichiscomplete,Scott,andstrict.

Thefollowingpropositionsaretrue:

(3)LetIbeanonemptysetandTbeaScotttopologicalaugmentationof∏(I →21 ).Then

thecarrierofT=thecarrierof∏(I →theSierpi´nskispace).

(4)LetL1,L2becompletelattices,T1beaScotttopologicalaugmentationofL1,T2beaScott

topologicalaugmentationofL2,hbeamapfromL1intoL2,andHbeamapfromT1intoT2.

Ifh=Handhisisomorphic,thenHisahomeomorphism.

(5)LetL1,L2becompletelattices,T1beaScotttopologicalaugmentationofL1,andT2be

aScotttopologicalaugmentationofL2.IfL1andL2areisomorphic,thenT1andT2are

homeomorphic.

(6)LetS,Tbenonemptytopologicalspaces.IfSisinjectiveandSandTarehomeomorphic,

thenTisinjective.

(7)LetL1,L2becompletelattices,T1beaScotttopologicalaugmentationofL1,andT2bea

ScotttopologicalaugmentationofL2.IfL1andL2areisomorphicandT1isinjective,thenT2

isinjective.

LetX,Ybenonemptytopologicalspaces.LetusobservethatXisatopologicalretractofYifandonlyif:

1ThisworkhasbeensupportedbyKBNGrant8T11C01812.

1cAssociationofMizarUsers

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