Topology Tools Department of Mathematics , Faculty of Science,University of Benghazi Taha Guma el turki E-mail: taha1978_2002@yahoo.com restart; with(combinat):with(networks):with(plottools): #(1)A procedure to check if a given collection over X is topology or not . CheckTopology:=proc(X,T) local i,O,CT,CuT; CT:={}; if `subset`(T,PowerSetof(X))=true then for O in T do CuT:={seq(O intersect T[i],i=1..nops(T)),seq(O union T[i],i=1..nops(T))}; CT:=CT union CuT; od; CT; CT:= CT union {{},X}; evalb(CT=T); if evalb(CT=T)=true then True ; else "False this collection does not represent a topology"; fi; else "False this collection is no a sub collection of the powerset"; fi; end: #(2) A procedure to obtain a relative topology on a subset of X. SubSpace:=proc(A,X,T) if (`subset`(A,X) and `subset`(T,PowerSetof(X))and CheckTopology(X,T))=True then map2(`intersect`,A,T); else "False check your entries"; fi; end: #(3) A procedure to check if a given point is a boundary point or not. CheckBoundary:=proc(x,A,X,T) local O,t,o,B; O:={}; B:={}; if (member(x,X)and`subset`(A,X)) then for t in T do if member(x,t) then O:= O union {t}; else O:=O; fi; od; O; for o in O do if (o intersect A )<> {} and (o intersect X minus A) <>{} then B:=B union {x}; else B:={};break; fi; od; B; else false; fi; B; if B={} then "It's not a boundary point of the given set"; else B:={x}; fi; end: #(4) A procedure to find all boundary points of a given Set. BoundaryPoints:=proc(A,X,T) local CB,x; CB:={}; if `subset`(A,X) then for x in X do if CheckBoundary(x,A,X,T)<>"It's not a boundary point of the given set" then CB:= CB union {x}; else CB:= CB; fi; od; CB; else false; fi; if CB={} then "The given set has no boundary points";else CB; fi; end: #(5) A procedure to check if a given point is interior point or not. CheckInterior:=proc(x,A,X,T) local O,In,eO,o; In:={}; eO:={}; if(`member`(x,X) and `subset`(A,X) ) then for O in T do if `member`(x,O) then eO:=eO union {O}; else eO:=eO; fi; od; for o in eO do if `subset`(o,A) then In:=In union {x}; else In:=In; fi; od; In; else false; fi; if In={} then "It's not an interior point of the given set"; else In:={x}; fi; end: #(6) A procedure to find all interior points of a given set. InteriorPoints:=proc(A,X,T) local x,ci,d,i; ci:={}; if `subset`(A,X)=true then for x in X do if CheckInterior(x,A,X,T)<>"It's not an interior point of the given set" then ci:=ci union {x}; else ci:=ci; fi; od; else print(false); fi; ci; if ci={} then "The given set has no interior points"; else ci; fi; end: #(7) A procedure to check if a given point is a closure point or not. CheckClosure:=proc(x,A,X,T) local c,o,cp,co; co:={}; cp:={}; if `member`(x,X) and `subset`(A,X) then for o in T do if `member`(x,o) then co:=co union {o} ;else co:=co; fi; od; co; for c in co do if c intersect A<>{} then cp:=cp union {x}; else cp:={};break; fi; od; cp; else false; fi; cp; if cp={} then "It's not a closure point of the given set"; else cp:={x}; fi; end: #(8) A procedure to find all closure points of a given set. ClosurePoints:=proc(A,X,T) local C,t,i,x; C:={}; if `subset`(A,X) then for x in X do if CheckClosure(x,A,X,T)<>"It's not a closure point of the given set" then C:=C union {x}; else C:=C; fi; od; C; else false; fi; if C={} then "The given set has no closure points";else C; fi; C; end: #(9)A procedure to check if a given point is an exterior point of a given subset or not. CheckExterior:=proc(x,A,X,T) local CX,O,ex,i; CX:= X minus A; ex:={}; if (member(x,X)and`subset`(A,X)) then for O in T do if `member`(x,O) and `subset`(O,CX) then ex:=ex union {x}; else ex:=ex; fi; od; ex; if ex={} then "It's not an exterior point of the given set " ; else ex:={x}; fi; else false; fi; end: #(10) A procedure to find all exterior points of a given subset. ExteriorPoints:=proc(A,X,T) local ext,x; ext:={}; if `subset`(A,X) then for x in X do if CheckExterior(x,A,X,T)<>"It's not an exterior point of the given set " then ext:=ext union {x}; else ext:=ext; fi; od; ext; if ext={} then"The given set has no exterior points"; else ext;fi; else false; fi; end: #(11) A procedure to check if a given point is a limit point or not. CheckLimit:=proc(x,A,X,T) local i,o,L,Omx,O; O:={}; L:={}; if member(x,X)= true then for i to nops(T) do if (member(x,T[i]))then O:= O union {T[i]}; else O:=O; fi; od; Omx:={seq(O[i] minus {x},i=1..nops(O))}; for o in Omx do if ((o intersect A) <> {}) then L:=L union {x}; else L:={}; break; fi; od; L; else false; fi; L; if L={} then "It's not a limit point of the given set ";else L:={x}; fi; end: #(12)A procedure to find all limit points of a given set of X. LimitPoints:=proc(A,X,T) local x,LI; LI:={}; if `subset`(A,X)=true then for x in X do if CheckLimit(x,A,X,T) <> "It's not a limit point of the given set " then LI:=LI union {x}; else LI:=LI; fi; od; LI; if LI={} then "The given set has no limit points";else LI; fi; else false ; fi; end: #(13)A procedure to check if a given point is isolated point or not. CheckIsolated:=proc(x,A,X,T) local o,iso; iso:={}; if (member(x,X)and`subset`(A,X) and `subset`(T,PowerSetof(X))) then for o in T do if o intersect A ={x} then iso:=iso union {x};break; else iso:=iso; fi; od; iso; if iso={} then "It's not an isolated point of the given set"; else iso:={x}; fi; else false; fi; end: #(14) A procedure to find all isolated points of a given subset. IsolatedPoints:=proc(A,X,T) local isol,x; isol:={}; if `subset`(A,X) then for x in X do if CheckIsolated(x,A,X,T)<>"It's not an isolated point of the given set" then isol:=isol union {x}; else isol:=isol; fi; od; isol; if isol={} then "The given set has no isolated points";else isol; fi; else false; fi; end: #(15)A procedure to find the closed sets of the topology. ClosedSets:=proc(X,T) {seq(X minus T[i],i=1..nops(T))}; end: #(16)A procedure to find the clopen sets of the topology. ClopenSets:=proc(X,T) local A,W; W:={}; for A in T do if `member`(X minus A,T)=true then W:=W union {A}; else W:=W; fi; od; W; end: #(17)A procedure to find the neither open nor closed setes in a given topology. NeitherOpenNorClosedSets:=proc(X,T) local A,B,C; if CheckTopology(X,T)=True then A:=PowerSetof(X) minus T; A; B:=A minus ClosedSets(X,T); B; C:=B minus ClopenSets(X,T); C; else "Please check your entries!!!";fi; if C={} then "There are no neither open nor closed sets of the given space"; else; C; fi; end: #(18)A procedure to check if a given topology is T0 or not. CheckT0:=proc(X,T) local x,y,O,test; if nops(X)=1 then print("True") ;else for x in X do for y in X minus{x} do for O in T do test:=evalb((member(x,O)and not(member(y,O)))or (member(y,O) and not(member(x,O)))); if test then break; fi; od: if not(test) then break;fi; od: if not(test) then break;fi; od: fi; test; end: #(19)A procedure to find all topologies over a given set. AllTopologies:=proc(X) local T,ALLTOPO; ALLTOPO:={}; for T in CollectionsUnionEmptyandFullSet(X) do if CheckTopology(X,T)=True then ALLTOPO:=ALLTOPO union {T};else ALLTOPO:=ALLTOPO; fi; od; ALLTOPO; end: #(20)A procedure to find the number of topologies over a given set. NumberofTopologies:=proc(X) print("There are",nops(AllTopologies(X)),"topologies over a set with",nops(X),"points"); end: #(21) A procedure to find all T0 topologies over a given set. AllT0Topologies:=proc(X) local i,T,T0S; T0S:={}; AllTopologies(X); if nops(X)=1 then T0S:=AllTopologies(X);else for T in AllTopologies(X) do if CheckT0(X,T)=true then T0S:= T0S union {T} ; else T0S:=T0S; fi; od; fi; T0S; end: #(22)A procedure to find the number of T0 topologies over a given set. NumberofT0Topologies:=proc(X) print("There are",nops(AllT0Topologies(X))," T0 topologies over a set with",nops(X),"points"); end: #(23)A procedure to generate a topology from a given subbasis S. GenerateTopologybySubBasis:=proc(X,S) local i,O,GT1,GT2,T,B; B:={}; T:={{}}; for O in S do GT1:={seq(O intersect S[i],i=1..nops(S))} ; B:=B union GT1; od; B:=B union {X}; for O in B do GT2:={seq(O union B[i],i=1..nops(B))}; B:= B union GT2; od; T union B; end: #(24) A procedure to find the minimal basic open set for a certain point x. MinimalBasic:= proc(x,X,T) local i,O,COUNT; COUNT:={}; if (member(x,X) and CheckTopology(X,T)=True) then for O in T do if member(x,O)then COUNT:=COUNT union {O}; else COUNT:=COUNT; fi; od; COUNT; COUNT[1]; else false; fi; end: #(25) A procedure to find the minimal basis of a given topological space. MinimalBasis:=proc(X,T) local x,minimalbasis; minimalbasis:={}; if `subset`(T,PowerSetof(X))then for x in X do minimalbasis:=minimalbasis union {MinimalBasic(x,X,T)}; od; minimalbasis; else print("Check the colllection !!"); fi; end: #(26)A procedure to close a given set under unions needed in other procedure. CloseUnions:=proc(T) local A,U;U:=T; for A in T do U:=U union map(`union`,U,A); od; if U=T then U; else CloseUnions(U); fi; end: #(27)A procedure to close agiven set under intersections needed in other procedure. CloseIntersections:=proc(T) local A,U;U:=T; for A in T do U:=U union map(`intersect`,U,A); od; if U=T then U; else CloseIntersections(U); fi; end: #(28)A procedure to check that if the topology is connected. CheckConnected:=proc(X,T) evalb(ClopenSets(X,T)={X,{}}); end: #(29)A procedure to find all connected topologies over a given set. AllConnectedSpaces:=proc(X) local C,c; C:={}; AllTopologies(X); for c in AllTopologies(X) do if CheckConnected(X,c)=true then C:= C union {c}; else; C:=C; fi; od; C; end: #(30)A procedure to find the number of connected spaces over a given set. NumberofConnectedSpaces:=proc(X) print(`There are`,nops(AllConnectedSpaces(X)),`connected spaces over a set with`,nops(X),`points`); end: #(31) A procedure to find the connected components of a given point. ConnectedComponents:=proc(x,X,T) local i,S,SK; SK:={}; S:=map2(`union`,{x},powerset(X)); for i to nops(S) do if CheckConnected(S[i],SubSpace(S[i],X,T)) then SK:=SK union S[i];fi; od; SK ; end: #(32) A procedure to find all connected components of a given space. AllConnectedComponentsTopology:=proc(X,T) local x,CC; CC:={}; for x in X do CC:=CC union {ConnectedComponents(x,X,T)}; od; CC; end: #(33)A procedure to check if a Topology is Totaly Disconnected. CheckTotalyDisconnected:=proc(X,T) local i; if CheckTopology(X,T)=True then for i to nops(X) do if not(ConnectedComponents(X[i],X,T)={X[i]}) then RETURN(false)fi; od; RETURN(true); else "Please check your enteries!!!";fi; end: #(34) A procedure to find all totaly disconnected space over a given set. AllTotalyDisconnected:=proc(X) local o,T;T:={}; AllTopologies(X); for o in AllTopologies(X) do if CheckTotalyDisconnected(X,o)=true then T:=T union {o};else T:=T; fi; od; end: #(35) A procedure to find the number of totaly disconnected spaces over a given set. NumberofTotalyDisconnectedSpaces:=proc(X) AllTotalyDisconnected(X); print(`There are`,nops(AllTotalyDisconnected(X )),`over a set with `,nops(X),`points`); end: #(36)A procedure to find all disconnected spces over a given set. AllDisconnectedSpaces:=proc(X) AllTopologies(X) minus AllConnectedSpaces(X); end: #(37)A procedure to find the number of disconnected spaces over a given set given set. NumberofDisconnectedSpaces:=proc(X) print(`There are`,nops(AllDisconnectedSpaces(X)),`disconnected spaces over a set with`,nops(X),`points`); end: #(38)A procedure to check if a given topology is weakly-dimensional or not. CheckWeaklyDimensional:=proc(X,T) if CheckTopology(X,T)=True then evalb(GenerateTopologybySubBasis(X,ClopenSets(X,T))=T) else "Please Check your entries"; fi; end: #(39)A procedure to find all waekaly-dimensional topology over a given set. AllWeaklyDimensionalSpaces:=proc(X) local W,o; W:={}; AllTopologies(X); for o in AllTopologies(X)do if CheckWeaklyDimensional(X,o)=true then W:=W union {o}; else W:=W; fi; od; end: #(40)A procedure to find the number of weakaly-dimensional topologies over a given set. NumberofWeaklyDimensionalSpaces:=proc(X) print(`There are`,nops(AllWeaklyDimensionalSpaces(X)),`weakly-dimensional topologies over a set with`,nops(X),`points`); end: #(41) A procedure to Generate Extremal Topology Over X by x,y. GenerateExtremalTopology:=proc(x,y,X) local o,c,t,e,px,py,M,txy; if `member`(x,X) and `member`(y,X) and x<>y then c:={};M:={};txy:={}; px:=PowerSetof(X minus {x}); for o in px do if `member`(y,o) then c:=c union {o}; else c:=c; fi; od; c; for e in c do txy:=e union {x}; M:=M union {txy}; od; px union M; else flase; fi; end: #(42)A proceure to find all extremal topologies over a finite set X. AllExtremalTopologies:=proc(n) local EX,x,y,pp,X; EX:={}; X:={seq(x[i],i=1..n)}; for x in X do for y in X minus {x} do EX:=EX union {GenerateExtremalTopology(x,y,X)}; od; od; EX; end: #(43)A Procdure to Find the Number of Extremal Topologies Over a Set with n Points. NumberofExtremalTopologies:=proc(n) print("There are ",nops(AllExtremalTopologies(n)),"extremal topologies over a set with",n,"points"); end: #(44) A procdure to find the Non-Trivial Minimal Topologies Over a Set with n Points. AllNonTrivialMinimalTopologies:=proc(n) local o,c,X,PX,T; c:={};T:={}; X:={seq(x[i],i=1..n)}; PX:=PowerSetof(X); for o in PX do c:=c union{{{},o,X}}; od; c minus {{{},X}}; end: #(45) A Procdure to Find the Number of Non-Trivial Minimal Topologies Over a Set with n Points. NumberofNonTrivialMinimalTopologies:=proc(n) print("There are",nops(AllNonTrivialMinimalTopologies(n)),"non trivial minimal topologies over a set with",n,"oints"); end: #(46)A Procdure to Find the Number of Open set in Generated Extremal Topologies. NumberofOpenSetsExtremal:=proc(x,y,X) print("Number of open setes is",nops(GenerateExtremalTopology(x,y,X))); end: #(47) A procedure to find the Cartesian product between two given sets. Expandcart:=proc(PX) local C,P,i; C:={}; P:=cartprod([seq([op(PX[i])],i=1..nops(PX))]); while not P[finished] do C:=C union{P[nextvalue]()}od; C; end: #(48)A procedure to Implementation of the inverse projections, this is needed in later procedures. InverseProj:=proc(k,S,PX) local i,O; for i to nops(PX) do if i=k then O[i]:=S else O[i]:=PX[i] fi; od; Expandcart([seq(O[i],i=1..nops(PX))]); end: #(49) A procedure to find subbasis for product topology using inverse projections. ProductBase:=proc(PX,PS) local i,k,S;S:={}; for k to nops (PX) do for i to nops(PS[k]) do S:=S union {InverseProj(k,PS[k][i],PX)}; od; od; S; end: #(50) A procedure to give alist of sets and alist of topologis,the product topology is found?. ProductTopology:=proc(PX,PT) GenerateTopologybySubBasis(Expandcart(PX),ProductBase(PX,PT)); end: #(51)A procedure to plot Hass digram over a given space. SetLattice:=proc(S) local a,b,G,os,i,j,lorder,P,L,T,k,tl,dincl; dincl:=proc(A,B,S) local C; if A=B then RETURN(false) fi; if not `subset`(A,B)then RETURN(false) fi; for C in S minus {A,B} do if `subset`(A,C)and `subset`(C,B)then RETURN(false) fi; od; RETURN(true); end: lorder:=proc(a,b)evalb(nops(a)<nops(b))end : new(G); os:=sort([op(S)],lorder); addvertex({seq(i,i=1..nops(S))},G); for i to nops(os) do for j to nops(os) do if dincl(os[i],os[j],S)then addedge([i,j],G)fi; od; od; k:=nops(os[1]); tl:=1; T:={}; for i to nops(os) do print(i=os[i]); if nops(os[i])=k then T:=T union {i}; else L[tl]:=T; T:={i}; tl:=tl+1; k:=nops(os[i]); fi; od; L[tl]:=T; P:=draw(Linear(seq([op(L[i])],i=1..tl)),G); print(rotate(P,Pi/2)); G; end: #(52)A procedure to finid the power set. PowerSetof:=proc(X) local Y; Y:=powerset(X); end: #(53)A procedure to find all proper sets over a given set. ProperSubsetsof:=proc(X) local Z; Z:=PowerSetof(X) minus{{},X}; end: #(54)A procedure to find proper sets collections over a given set. CollectionsbyProperSubsets:=proc(X) local W; W:=PowerSetof(ProperSubsetsof(X)); end: #(55)A procedure proper sets collections over a given set union {{},Full Set}. CollectionsUnionEmptyandFullSet:=proc(X) {seq(w union{{},X},w=CollectionsbyProperSubsets(X))}; end: #(56) A procedure to apply a permutation p of the elements of a space to the sets in a topology t is needed in later preocedure. newjob:=proc(t,p) local u: {seq(subs({seq(X[i]=p[i],i=1..nops(X))} ,u) ,u=t)}: end: #(57)A procedure to compare sizes between a given topologies is needed in the following topology. bigger:=proc(t1,t2) if nops(t1) < nops(t2) then true else false fi: end: #(58) A procedure to Check if two topologies are homeomoprhic or not. IsHomeomorphic:=proc(t1,t2,X) local answer, P,p; if (CheckTopology(X,t1)="False this collection does not represent a topology" or CheckTopology(X,t2)="False this collection does not represent a topology" or not`subset`(t1,PowerSetof(X)) or not`subset`(t2,PowerSetof(X))) then "There are some mistaiks in your inputs please check it!"; else p:=permute(X); #we can first check for some trivial invariants, such as. if nops(t1) <> nops(t2) then return(false) fi: answer:=false: for p in P do if {op(newjob(t1,p))}={op(t2)} then answer:=true: break: fi:od: #Note that we have to compare _sets_ rather than _lists_!. answer: fi; end: #(59)A procedure to find the proper disjoint closed sets. DisjointProperClosedSets:=proc(X,T) local c,p,o,SC,CC; SC:={}; CC:=ClosedSets(X,T) minus {{},X}; for c in CC do for p in CC minus c do if c intersect p ={} then SC:=SC union {c,p}; else SC:=SC; fi; od;od; SC; end: #(60)A procedure to find the disjoint proper open sets of a given topological space. DisjointProperOpenSets:=proc(X,T) local o,m,D,PO; D:={}; PO:=T minus {{},X}; for o in PO do for m in PO minus {o} do if o intersect m ={} then D:=D union {o,m};else D:=D; fi;od;od; D; end: #(61) A procedure to find disoint closed sets in pairs {,}. DisjointClosedinPairs:=proc(X,T) local f,c,C,M; C:={}; M:=ClosedSets(X,T)minus {{}}; for f in M do for c in M minus {f} do if c intersect f={} then C:=C union {{c,f}}; else C:=C; fi; od; od; C; end: #(62) A procedure to check if a given space is normal or not. CheckNormal:=proc(X,T) local o,p,c,s,F; F:={}; if nops(T)=2 or nops(T)=3 or DisjointClosedinPairs(X,T)={} then true; else for o in DisjointProperOpenSets(X,T) do for c in DisjointProperClosedSets(X,T) do for p in DisjointProperOpenSets(X,T) minus {o} do for s in DisjointProperClosedSets(X,T) minus {c} do if`subset`(c,o)and `subset`(s,p) then F:=((F union {o})union {p}); else F:=F; fi; od; od; od; od; F; if F={}then false; else true; fi; fi; end: #(63)A procedure to find all Normal spaces over a given set. AllNormalSpaces:=proc(X) local o,N; N:={}; AllTopologies(X); for o in AllTopologies(X) do if CheckNormal(X,o)=true then N:=N union {o}; else N:=N; fi; od; N; end: #(64)A procedure to find the number of normal spaces over a given set. NumberofNormalSpaces:=proc(X) AllNormalSpaces(X); print(`There are`,nops(AllNormalSpaces(X)),`Normal spaces over a set with`,nops(X),`points`); end: #(65) A procedure to find non normal Spaces over a given set. NonNormalSpaces:=proc(X) AllTopologies(X) minus AllNormalSpaces(X); end: #Implementations; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= X:={a,b,c}; PCVJImFHNiJJImJHRiRJImNHRiQ= T:={{}, {a}, {a, b}, {a, c}, {a, b, c}}; CheckTopology(X,T); PCc8IjwjSSJhRzYiPCRGJUkiYkdGJjwkRiVJImNHRiY8JUYlRihGKg== SSVUcnVlRzYi A:={b,c}; PCRJImJHNiJJImNHRiQ= SubSpace(A,X,T); PCY8IjwjSSJiRzYiPCNJImNHRiY8JEYlRig= CheckBoundary(a,A,X,T); UUtJdCdzfm5vdH5hfmJvdW5kYXJ5fnBvaW50fm9mfnRoZX5naXZlbn5zZXQ2Ig== BoundaryPoints(A,X,T); PCRJImJHNiJJImNHRiQ= CheckInterior(c,A,X,T); UUxJdCdzfm5vdH5hbn5pbnRlcmlvcn5wb2ludH5vZn50aGV+Z2l2ZW5+c2V0NiI= InteriorPoints(A,X,T); UUVUaGV+Z2l2ZW5+c2V0fmhhc35ub35pbnRlcmlvcn5wb2ludHM2Ig== CheckClosure(c,A,X,T); PCNJImNHNiI= ClosurePoints(A,X,T); PCRJImJHNiJJImNHRiQ= CheckExterior(b,A,X,T); UU1JdCdzfm5vdH5hbn5leHRlcmlvcn5wb2ludH5vZn50aGV+Z2l2ZW5+c2V0fjYi ExteriorPoints(A,X,T); PCNJImFHNiI= CheckLimit(c,A,X,T); UUlJdCdzfm5vdH5hfmxpbWl0fnBvaW50fm9mfnRoZX5naXZlbn5zZXR+NiI= LimitPoints(A,X,T); UUJUaGV+Z2l2ZW5+c2V0fmhhc35ub35saW1pdH5wb2ludHM2Ig== CheckIsolated(b,A,X,T); PCNJImJHNiI= IsolatedPoints(A,X,T); PCRJImJHNiJJImNHRiQ= ClosedSets(X,T); PCc8IjwjSSJiRzYiPCNJImNHRiY8JEYlRig8JUkiYUdGJkYlRig= ClopenSets(X,T); PCQ8IjwlSSJhRzYiSSJiR0YmSSJjR0Ym NeitherOpenNorClosedSets(X,T); UWduVGhlcmV+YXJlfm5vfm5laXRoZXJ+b3Blbn5ub3J+Y2xvc2VkfnNldHN+b2Z+dGhlfmdpdmVufnNwYWNlNiI= CheckT0(X,T); SSV0cnVlRyUqcHJvdGVjdGVkRw== AllTopologies(X); PD88JDwiPCVJImFHNiJJImJHRidJImNHRic8JUYkPCNGJkYlPCVGJDwjRihGJTwlRiQ8I0YpRiU8JUYkPCRGJkYoRiU8JUYkPCRGJkYpRiU8JUYkPCRGKEYpRiU8JkYkRitGMUYlPCZGJEYrRjNGJTwmRiRGK0Y1RiU8JkYkRi1GMUYlPCZGJEYtRjNGJTwmRiRGLUY1RiU8JkYkRi9GMUYlPCZGJEYvRjNGJTwmRiRGL0Y1RiU8J0YkRitGLUYxRiU8J0YkRitGL0YzRiU8J0YkRitGMUYzRiU8J0YkRi1GL0Y1RiU8J0YkRi1GMUY1RiU8J0YkRi9GM0Y1RiU8KEYkRitGLUYxRjNGJTwoRiRGK0YtRjFGNUYlPChGJEYrRi9GMUYzRiU8KEYkRitGL0YzRjVGJTwoRiRGLUYvRjFGNUYlPChGJEYtRi9GM0Y1RiU8KkYkRitGLUYvRjFGM0Y1RiU= NumberofTopologies(X); NidRKlRoZXJlfmFyZTYiIiNIUTt0b3BvbG9naWVzfm92ZXJ+YX5zZXR+d2l0aEYkIiIkUSdwb2ludHNGJA== AllT0Topologies(X); PDU8JjwiPCNJImFHNiI8JEYmSSJiR0YnPCVGJkYpSSJjR0YnPCZGJEYlPCRGJkYrRio8JkYkPCNGKUYoRio8JkYkRi88JEYpRitGKjwmRiQ8I0YrRi1GKjwmRiRGM0YxRio8J0YkRiVGL0YoRio8J0YkRiVGM0YtRio8J0YkRiVGKEYtRio8J0YkRi9GM0YxRio8J0YkRi9GKEYxRio8J0YkRjNGLUYxRio8KEYkRiVGL0YoRi1GKjwoRiRGJUYvRihGMUYqPChGJEYlRjNGKEYtRio8KEYkRiVGM0YtRjFGKjwoRiRGL0YzRihGMUYqPChGJEYvRjNGLUYxRio8KkYkRiVGL0YzRihGLUYxRio= NumberofT0Topologies(X); NidRKlRoZXJlfmFyZTYiIiM+UT9+VDB+dG9wb2xvZ2llc35vdmVyfmF+c2V0fndpdGhGJCIiJFEncG9pbnRzRiQ= K:={{a}}; PCM8I0kiYUc2Ig== GenerateTopologybySubBasis(X,K); PCU8IjwjSSJhRzYiPCVGJUkiYkdGJkkiY0dGJg== MinimalBasic(a,X,T); PCNJImFHNiI= MinimalBasis(X,T); PCU8I0kiYUc2IjwkRiRJImJHRiU8JEYkSSJjR0Yl CloseUnions(T); PCc8IjwjSSJhRzYiPCRGJUkiYkdGJjwkRiVJImNHRiY8JUYlRihGKg== CloseIntersections(T); PCc8IjwjSSJhRzYiPCRGJUkiYkdGJjwkRiVJImNHRiY8JUYlRihGKg== CheckConnected(X,T); SSV0cnVlRyUqcHJvdGVjdGVkRw== AllConnectedSpaces(X); PDU8JDwiPCVJImFHNiJJImJHRidJImNHRic8JUYkPCNGJkYlPCVGJDwjRihGJTwlRiQ8I0YpRiU8JUYkPCRGJkYoRiU8JUYkPCRGJkYpRiU8JUYkPCRGKEYpRiU8JkYkRitGMUYlPCZGJEYrRjNGJTwmRiRGLUYxRiU8JkYkRi1GNUYlPCZGJEYvRjNGJTwmRiRGL0Y1RiU8J0YkRitGLUYxRiU8J0YkRitGL0YzRiU8J0YkRitGMUYzRiU8J0YkRi1GL0Y1RiU8J0YkRi1GMUY1RiU8J0YkRi9GM0Y1RiU= NumberofConnectedSpaces(X); NidJKlRoZXJlfmFyZUc2IiIjPklBY29ubmVjdGVkfnNwYWNlc35vdmVyfmF+c2V0fndpdGhHRiQiIiRJJ3BvaW50c0dGJA== ConnectedComponents(a,X,T); PCVJImFHNiJJImJHRiRJImNHRiQ= AllConnectedComponentsTopology(X,T); PCM8JUkiYUc2IkkiYkc2IkkiY0c2Ig== CheckTotalyDisconnected(X,T); SSZmYWxzZUclKnByb3RlY3RlZEc= AllTotalyDisconnected(X); PCM8KjwiPCNJImFHNiI8I0kiYkc2IjwjSSJjRzYiPCRJImFHNiJJImJHNiI8JEkiYUc2IkkiY0c2IjwkSSJiRzYiSSJjRzYiPCVJImFHNiJJImJHNiJJImNHNiI= NumberofTotalyDisconnectedSpaces(X); NidJKlRoZXJlfmFyZUc2IiIiIkkxb3Zlcn5hfnNldH53aXRofkdGJCIiJEkncG9pbnRzR0Yk AllDisconnectedSpaces(X); PCw8JjwiPCNJImFHNiI8JEkiYkdGJ0kiY0dGJzwlRiZGKUYqPCZGJDwjRik8JEYmRipGKzwmRiQ8I0YqPCRGJkYpRis8KEYkRiVGLUYxRi5GKzwoRiRGJUYtRjFGKEYrPChGJEYlRjBGMUYuRis8KEYkRiVGMEYuRihGKzwoRiRGLUYwRjFGKEYrPChGJEYtRjBGLkYoRis8KkYkRiVGLUYwRjFGLkYoRis= NumberofDisconnectedSpaces(X); NidJKlRoZXJlfmFyZUc2IiIjNUlEZGlzY29ubmVjdGVkfnNwYWNlc35vdmVyfmF+c2V0fndpdGhHRiQiIiRJJ3BvaW50c0dGJA== CheckWeaklyDimensional(X,T); SSZmYWxzZUclKnByb3RlY3RlZEc= AllWeaklyDimensionalSpaces(X); PCc8JDwiPCVJImFHNiJJImJHRidJImNHRic8JkYkPCNGJjwkRihGKUYlPCZGJDwjRig8JEYmRilGJTwmRiQ8I0YpPCRGJkYoRiU8KkYkRitGLkYxRjJGL0YsRiU= NumberofWeaklyDimensionalSpaces(X); NidJKlRoZXJlfmFyZUc2IiIiJklOd2Vha2x5LWRpbWVuc2lvbmFsfnRvcG9sb2dpZXN+b3Zlcn5hfnNldH53aXRoR0YkIiIkSSdwb2ludHNHRiQ= GenerateExtremalTopology(a,c, X); PCg8IjwjSSJiRzYiPCNJImNHRiY8JEkiYUdGJkYoPCRGJUYoPCVGKkYlRig= AllExtremalTopologies(3); PCg8KDwiPCMmSSJ4RzYjL0knX2xvY2FsRyUqcHJvdGVjdGVkR0ksX242NDIyMDY0NjRHNiI2IyIiIjwjJkYnNiMiIiM8JEYmRjE8JEYmJkYnNiMiIiQ8JUYmRjFGNjwoRiRGJUYwRjQ8JEYxRjZGOTwoRiRGJTwjRjZGNEY1Rjk8KEYkRiVGPUY1RjtGOTwoRiRGMEY9RjRGO0Y5PChGJEYwRj1GNUY7Rjk= NumberofExtremalTopologies(3); NidRK1RoZXJlfmFyZX42IiIiJ1FEZXh0cmVtYWx+dG9wb2xvZ2llc35vdmVyfmF+c2V0fndpdGhGJCIiJFEncG9pbnRzRiQ= AllNonTrivialMinimalTopologies(3); PCg8JTwiPCMmSSJ4RzYiNiMiIiI8JUYmJkYnNiMiIiMmRic2IyIiJDwlRiQ8I0YsRis8JUYkPCNGL0YrPCVGJDwkRiZGLEYrPCVGJDwkRiZGL0YrPCVGJDwkRixGL0Yr NumberofNonTrivialMinimalTopologies(5); NidRKlRoZXJlfmFyZTYiIiNJUU9ub25+dHJpdmlhbH5taW5pbWFsfnRvcG9sb2dpZXN+b3Zlcn5hfnNldH53aXRoRiQiIiZRJm9pbnRzRiQ= NumberofOpenSetsExtremal(a,c,X); NiRROE51bWJlcn5vZn5vcGVufnNldGVzfmlzNiIiIic= Y:={0,1}; PCQiIiEiIiI= S:={{},{0},Y}; PCU8IjwjIiIhPCRGJSIiIg== CheckTopology(Y,S); SSVUcnVlRzYi XY:=Expandcart([X,Y]); PCg3JEkiYUc2IiIiITckRiQiIiI3JEkiYkdGJUYmNyRGKkYoNyRJImNHRiVGJjckRi1GKA== TXY:=ProductTopology([X,Y],[T,S]);nops(TXY); 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Z:={0,1}; T1:={{},{0},Z}; T2:={{},{0},Z}; PCQiIiEiIiI= PCU8IjwjIiIhPCRGJSIiIg== PCU8IjwjIiIhPCRGJSIiIg== IsHomeomorphic(T1,T2,Z); SSV0cnVlRyUqcHJvdGVjdGVkRw== DisjointProperClosedSets(X,T); PCQ8I0kiYkc2IjwjSSJjR0Yl DisjointProperOpenSets(X,T); PCI= CheckNormal(X,T); SSZmYWxzZUclKnByb3RlY3RlZEc= AllNormalSpaces(X,T); PDw8JDwiPCVJImFHNiJJImJHRidJImNHRic8JUYkPCNGJkYlPCVGJDwjRihGJTwlRiQ8I0YpRiU8JUYkPCRGJkYoRiU8JUYkPCRGJkYpRiU8JUYkPCRGKEYpRiU8JkYkRitGMUYlPCZGJEYrRjNGJTwmRiRGK0Y1RiU8JkYkRi1GMUYlPCZGJEYtRjNGJTwmRiRGLUY1RiU8JkYkRi9GMUYlPCZGJEYvRjNGJTwmRiRGL0Y1RiU8J0YkRitGLUYxRiU8J0YkRitGL0YzRiU8J0YkRi1GL0Y1RiU8KEYkRitGLUYxRjNGJTwoRiRGK0YtRjFGNUYlPChGJEYrRi9GMUYzRiU8KEYkRitGL0YzRjVGJTwoRiRGLUYvRjFGNUYlPChGJEYtRi9GM0Y1RiU8KkYkRitGLUYvRjFGM0Y1RiU= NumberofNormalSpaces(X); NidJKlRoZXJlfmFyZUc2IiIjRUk+Tm9ybWFsfnNwYWNlc35vdmVyfmF+c2V0fndpdGhHRiQiIiRJJ3BvaW50c0dGJA== NonNormalSpaces(X); PCU8JzwiPCNJImFHNiI8JEYmSSJiR0YnPCRGJkkiY0dGJzwlRiZGKUYrPCdGJDwjRilGKDwkRilGK0YsPCdGJDwjRitGKkYvRiw= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=