Formal Topological Spaces

Transcription

1 FORMALIZED MATHEMATICS Volume 9, Number 3, 2001 University of Białystok Formal Topological Spaces Gang Liu Tokyo University of Mercantile Marine Yasushi Fuwa Shinshu University Nagano Masayoshi Eguchi Tokyo University of Mercantile Marine SummaryThisarticleisdividedintotwopartsInthefirstpart,weprove some useful theorems on finite topological spaces In the second part, in order to considerafamilyofneighborhoodstoapointinaspace,weextendthenotionof finite topological space and define a new topological space, which we call formal topological space We show the relation between formal topological space struct (FMT Space Str) and the TopStruct by giving a function(neighsp) And the following notions are introduced in formal topological spaces: boundary, closure, interior, isolated point, connected point, open set and close set, then some basic facts concerning them are proved We will discuss the relation between the formal topological space and the finite topological space in future work MML Identifier: FINTOPO2 Thepapers[5],[3],[2],[1],[6],and[4]providethenotationandterminologyfor this paper 1 Some Useful Theorems on Finite Topological Spaces Inthispaper F 1 denotesanonemptyfinitetopologyspaceand Adenotesa subsetofthecarrierof F 1 The following propositions are true: (1) Let F 1 beanonemptyfinitetopologyspaceand A, Bbesubsetsofthe carrierof F 1 If A B,then A i B i (2) A δ = A b (A i ) c 537 c 2001 University of Białystok ISSN

2 538 gangliuetal (3) A δ = A b \ A i (4) If A c isopen,then Aisclosed (5) If A c isclosed,then Aisopen carrierof F 1,let ybeanelementofthecarrierof F 1,andlet Abeasubsetof thecarrierof F 1 ThefunctorP 1 (x,y,a)yieldsanelementofbooleanandis defined by: { true,if y U(x)and y A, (Def1) P 1 (x,y, A) = false, otherwise carrierof F 1,let ybeanelementofthecarrierof F 1,andlet Abeasubset ofthecarrierof F 1 ThefunctorP 2 (x,y, A)yieldinganelementofBooleanis defined as follows: { true,if y U(x)and y A c, (Def2) P 2 (x,y, A) = false, otherwise We now state three propositions: (6) Let x, ybeelementsofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 ThenP 1 (x,y, A) =trueifandonlyif y U(x)and y A (7) Let x, ybeelementsofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 ThenP 2 (x,y, A) =trueifandonlyif y U(x)and y A c (8) Let xbeanelementofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 Then x A δ ifandonlyifthereexistelements y 1, y 2 ofthecarrier of F 1 suchthatp 1 (x,y 1,A) =trueandp 2 (x,y 2,A) =true carrierof F 1,andlet ybeanelementofthecarrierof F 1 ThefunctorP 0 (x,y) yielding an element of Boolean is defined as follows: { true,if y U(x), (Def3) P 0 (x,y) = false, otherwise We now state three propositions: (9) Forallelements x, yofthecarrierof F 1 holdsp 0 (x,y) =trueiff y U(x) (10) Let xbeanelementofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 Then x A i ifandonlyifforeveryelement yofthecarrierof F 1 suchthatp 0 (x,y) =trueholdsp 1 (x,y, A) =true (11) Let xbeanelementofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 Then x A b ifandonlyifthereexistsanelement y 1 ofthecarrier of F 1 suchthatp 1 (x,y 1,A) =true carrierof F 1,andlet Abeasubsetofthecarrierof F 1 ThefunctorP A (x,a) yielding an element of Boolean is defined as follows:

3 formal topological spaces 539 { true,if x A, (Def4) P A (x,a) = false, otherwise One can prove the following three propositions: (12) Let xbeanelementofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 ThenP A (x,a) =trueifandonlyif x A (13) Let xbeanelementofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 Then x A δ i ifandonlyifthefollowingconditionsaresatisfied: (i) thereexistelements y 1, y 2 ofthecarrierof F 1 suchthatp 1 (x,y 1,A) = trueandp 2 (x, y 2,A) =true,and (ii) P A (x,a) =true (14) Let xbeanelementofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 Then x A δ o ifandonlyifthefollowingconditionsaresatisfied: (i) thereexistelements y 1, y 2 ofthecarrierof F 1 suchthatp 1 (x,y 1,A) = trueandp 2 (x, y 2,A) =true,and (ii) P A (x,a) =false carrierof F 1,andlet ybeanelementofthecarrierof F 1 ThefunctorP e (x,y) yielding an element of Boolean is defined by: { true,if x = y, (Def5) P e (x,y) = false, otherwise The following four propositions are true: (15) Forallelements x, yofthecarrierof F 1 holdsp e (x,y) =trueiff x = y (16) Let xbeanelementofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 Then x A s ifandonlyifthefollowingconditionsaresatisfied: (i) P A (x,a) =true,and (ii) itisnottruethatthereexistsanelement yofthecarrierof F 1 such thatp 1 (x,y, A) =trueandp e (x,y) =false (17) Let xbeanelementofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 Then x A n ifandonlyifthefollowingconditionsaresatisfied: (i) P A (x,a) =true,and (ii) thereexistsanelement yofthecarrierof F 1 suchthatp 1 (x,y,a) =true andp e (x,y) =false (18) Let xbeanelementofthecarrierof F 1 and Abeasubsetofthecarrier of F 1 Then x A f ifandonlyifthereexistsanelement yofthecarrier of F 1 suchthatp A (y, A) =trueandp 0 (y,x) =true

4 540 gangliuetal 2 Formal Topological Spaces We introduce formal topological spaces which are extensions of 1-sorted structure and are systems a carrier, a Neighbour-map, wherethecarrierisasetandtheneighbour-mapisafunctionfromthecarrier into 2 2the carrier Letusobservethatthereexistsaformaltopologicalspacewhichisnon empty and strict Inthesequel T isanonemptytopologicalstructure, F 2 isanonempty formaltopologicalspace,and xisanelementofthecarrierof F 2 Letusconsider F 2 andlet xbeanelementofthecarrierof F 2 Thefunctor U F (x)yieldingasubsetof 2 thecarrierof F 2 isdefinedasfollows: (Def6) U F (x) =(theneighbour-mapof F 2 )(x) Next we state the proposition (19) Let F 2 beanonemptyformaltopologicalspaceand xbeanelementof thecarrierof F 2 Then U F (x) =(theneighbour-mapof F 2 )(x) Letusconsider TThefunctorNeighSpTyieldinganonemptystrictformal topological space is defined by the conditions(def 7) (Def7)(i) ThecarrierofNeighSpT =thecarrierof T,and (ii) foreverypoint xofneighsptholds U F (x) ={V ;Vrangesoversubsets of T: V thetopologyof T x V} Inthesequel A, B, W, Vdenotesubsetsofthecarrierof F 2 Let I 1 beanonemptyformaltopologicalspacewesaythat I 1 isfilledif andonlyif: (Def8) Foreveryelement xofthecarrierof I 1 andforeverysubset Dofthe carrierof I 1 suchthat D U F (x)holds x D Letusconsider F 2 andletusconsider AThefunctor A Fδ yieldingasubset ofthecarrierof F 2 isdefinedasfollows: (Def9) A Fδ ={x : W (W U F(x) W A W A c )} The following proposition is true (20) x A Fδ iffforevery Wsuchthat W U F (x)holds W A and W A c Letusconsider F 2 andletusconsider AThefunctor A F byieldingasubset ofthecarrierof F 2 isdefinedasfollows: (Def10) A F b ={x : W (W U F(x) W A )} One can prove the following proposition (21) x A F biffforevery Wsuchthat W U F (x)holds W A

5 formal topological spaces 541 Letusconsider F 2 andletusconsider AThefunctor A F i yieldingasubset ofthecarrierof F 2 isdefinedasfollows: (Def11) A F i ={x : V (V U F(x) V A)} Next we state the proposition (22) x A F i iffthereexists Vsuchthat V U F (x)and V A Letusconsider F 2 andletusconsider AThefunctor A F s yieldsasubsetof thecarrierof F 2 andisdefinedby: (Def12) A F s ={x : x A V (V U F(x) (V\{x}) A = )} One can prove the following proposition (23) x A Fs iff x Aandthereexists V suchthat V U F (x)and (V\ {x}) A = Letusconsider F 2 andletusconsider AThefunctor A F on yieldsasubset ofthecarrierof F 2 andisdefinedby: (Def13) A Fon = A\A Fs We now state a number of propositions: (24) x A F on iff x Aandforevery V suchthat V U F (x)holds (V\ {x}) A (25) Let F 2 beanonemptyformaltopologicalspaceand A, Bbesubsetsof thecarrierof F 2 If A B,then A F b B F b (26) Let F 2 beanonemptyformaltopologicalspaceand A, Bbesubsetsof thecarrierof F 2 If A B,then A F i B F i (27) A F b B F b A B F b (28) A B F b A F b B F b (29) A F i B F i A B F i (30) A B F i A F i B F i (31) Let F 2 beanonemptyformaltopologicalspacethenthefollowing statements are equivalent (i) foreveryelement xofthecarrierof F 2 andforallsubsets V 1, V 2 ofthe carrierof F 2 suchthat V 1 U F (x)and V 2 U F (x)thereexistsasubset Wofthecarrierof F 2 suchthat W U F (x)and W V 1 V 2, (ii) forallsubsets A, Bofthecarrierof F 2 holds A B F b = A F b B F b (32) Let F 2 beanonemptyformaltopologicalspacethenthefollowing statements are equivalent (i) foreveryelement xofthecarrierof F 2 andforallsubsets V 1, V 2 ofthe carrierof F 2 suchthat V 1 U F (x)and V 2 U F (x)thereexistsasubset Wofthecarrierof F 2 suchthat W U F (x)and W V 1 V 2, (ii) forallsubsets A, Bofthecarrierof F 2 holds A F i B F i = A B F i (33) Let F 2 beanonemptyformaltopologicalspaceand A, Bbesubsets ofthecarrierof F 2 Supposethatforeveryelement xofthecarrierof

6 542 gangliuetal F 2 andforallsubsets V 1, V 2 ofthecarrierof F 2 suchthat V 1 U F (x) and V 2 U F (x)thereexistsasubset W ofthecarrierof F 2 suchthat W U F (x)and W V 1 V 2 Then A B Fδ A Fδ B Fδ (34) Let F 2 beanonemptyformaltopologicalspacesuppose F 2 isfilled Supposethatforallsubsets A, Bofthecarrierof F 2 holds A B Fδ = A Fδ B Fδ Let xbeanelementofthecarrierof F 2 and V 1, V 2 besubsetsof thecarrierof F 2 Suppose V 1 U F (x)and V 2 U F (x)thenthereexists asubset Wofthecarrierof F 2 suchthat W U F (x)and W V 1 V 2 (35) Let xbeanelementofthecarrierof F 2 and Abeasubsetofthecarrier of F 2 Then x A F s ifandonlyifthefollowingconditionsaresatisfied: (i) x A,and (ii) x / A\{x} F b (36) Let F 2 beanonemptyformaltopologicalspacethen F 2 isfilledifand onlyifforeverysubset Aofthecarrierof F 2 holds A A F b (37) Let F 2 beanonemptyformaltopologicalspacethen F 2 isfilledifand onlyifforeverysubset Aofthecarrierof F 2 holds A F i A (38) (A cf b ) c = A F i (39) (A cf i ) c = A F b (40) A Fδ = A F b A cf b (41) A Fδ = A F b (A F i ) c (42) A Fδ = A cfδ (43) A Fδ = A F b\ A F i Letusconsider F 2 andletusconsider AThefunctor A Fδ i yieldsasubset ofthecarrierof F 2 andisdefinedby: (Def14) A Fδ i = A A Fδ Thefunctor A Fδ o yieldsasubsetofthecarrierof F 2 andisdefinedby: (Def15) A Fδ o = A c A Fδ The following proposition is true (44) A Fδ = A Fδ i A Fδ o Letusconsider F 2 andlet Gbeasubsetofthecarrierof F 2 Wesaythat G isopenifandonlyif: (Def16) G = G F i Wesaythat Gisclosedifandonlyif: (Def17) G = G F b Next we state four propositions: (45) If A c isopen,then Aisclosed (46) If A c isclosed,then Aisopen

7 formal topological spaces 543 (47) Let F 2 beanonemptyformaltopologicalspaceand A, Bbesubsetsof thecarrierof F 2 Suppose F 2 isfilledsupposethatforeveryelement xof thecarrierof F 2 holds{x} U F (x)then A B F b = A F b B F b (48) Let F 2 beanonemptyformaltopologicalspaceand A, Bbesubsetsof thecarrierof F 2 Suppose F 2 isfilledsupposethatforeveryelement xof thecarrierof F 2 holds{x} U F (x)then A F i B F i = A B F i References [1] Czesław Byliński Functions from a set to a set Formalized Mathematics, 1(1): , 1990 [2] Hiroshi Imura and Masayoshi Eguchi Finite topological spaces Formalized Mathematics, 3(2): , 1992 [3] Andrzej Trybulec Domains and their Cartesian products Formalized Mathematics, 1(1): , 1990 [4] Zinaida Trybulec Properties of subsets Formalized Mathematics, 1(1):67 71, 1990 [5] Zinaida Trybulec and Halina Święczkowska Boolean properties of sets Formalized Mathematics, 1(1):17 23, 1990 [6] Edmund Woronowicz Many argument relations Formalized Mathematics, 1(4): , 1990 Received October 13, 2000