Maple in Finite Topological Spaces \342\200\223Special PointsKahtan H.Alzubaidy, Taha Guma Elturki Department of Mathematics , Faculty of Science ,University of Benghazi e-mails : kahtanalzubaidy@yahoo.com , taha1978_2002@yahoo.com
<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>The special points of a set in a topological space are limit points , closure points , interior points , boundary points ,exterior points and isolated points. Except limit points and isolated points the other special points can be computed by implementing simple formulas.For limit points we have to resort to the very definition to find them. On the other hand all special points can be derived from limit points . We have found computer procedures to compute the limit points of a set in finite space . Upon these procedures we have created other procedures to find the other special points . The software used is Maple.15 . Some ready-made procedures are also used.Let X be a finite topological space and A is a sub set of X . If the limit points , closure , interior , boundary , exterior and isolated points of A are denoted by LimitPoints(A) , ClosurePoints(A) , BoundaryPoints(A) , InteriorPoints(A) , ExteriorPoints(A) and IsolatedPoints(A) respectively, then we have :ClosurePoints(A) = A U LimitPoints(A) .BoundaryPoints(A) = ClosurePoints(A) \342\210\251 ClosurePoints(X-A) .InteriorPoints(A) = ClosurePoints(A) - BoundaryPoints(A) .ExteriorPoints(A) = InteriorPoints(X - A) .IsolatedPoints(A) = A - LimitPoints(A) .References
<Text-field style="Heading 1" layout="Heading 1">References</Text-field>[1] Dider Deses : Math-Page http : // student.vub.ac.be./~diddesen/math.html (2001). [2] www.maplesoft.com/applications/view.aspx?SID=4122&view=html (2001).
<Text-field style="Heading 1" layout="Heading 1">Procedure Definitions</Text-field>restart;with(combinat):
NiI=
#(1)A procedure to find all possible intersections of a given subbasis(S);
Basis:=proc(S)
local s,U,B;
U:=S;
for s in S do
U:=U union map(`intersect`,U,s);
od;
B:=U union {X}; #to add the empty intersection;
end:
#(2)A procedure used to generate a topology(T) from basis(B);
Topology:=proc(B)
local b,U,t;
U:=Basis(S);
for b in Basis(S) do
U:=U union map(`union`,U,b);
od;
t:=U union {{}};#to add the empty union;
end:
#(3)A procedures to check if (T) is a topology or not;
CloseIntersection:=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 CloseIntersection(U); fi;
end:
CloseUnion:=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 CloseUnion(U); fi;
end:
IsTopology:=proc(T)
CloseIntersection(T)=T and CloseUnion(T)=T and member({},T) and member(X,T)and `subset`(T,powerset(X));
end:
NiI=#(1)A procedure to check if a given point is a limit point or not;
IsLimitPoint:=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;
end:
#(2)A procedure to find limit points of a given subset of X;
LimitPoints:=proc(A,X,T)
local x,LI;
LI:={};
if `subset`(A,X)= true then
for x in X do
if IsLimitPoint(x,A,X,T) <> {} then LI:=LI union {x};
else LI:=LI;
fi;
od;
LI;
else false ;
fi;
end:
#(3)A procedure to find the closure points of a given subset of X ;
ClosurePoints:=proc(A,X,T)
A union LimitPoints(A,X,T);
end:
#(4)A procedure to find the boundary points of a given subset of X;
BoundaryPoints:=proc(A,X,T)
ClosurePoints(A,X,T) intersect ClosurePoints(X minus A,X,T);
end:
#(5)A procedure to find the interior points of a given subset of X;
InteriorPoints:=proc(A,X,T);
ClosurePoints(A,X,T) minus BoundaryPoints(A,X,T);
end:
#(6) A procedure to find the exterior points of a given subset of X ;
ExteriorPoints:=proc(A,X,T)
InteriorPoints(X minus A,X,T);
end:
#(7)A procedure to find the isolated points of a given subset of X;
IsolatedPoints:=proc(A,X,T)
A minus LimitPoints(A,X,T);
end:
#(8) A Procedure to find the isolated points of a given subset of X by definition ;
IsolatedPoints2:=proc(A,X,T)
local O,x,iso;
iso:={};
for x in X do
for O in T do
if( member(x,O) and O intersect A ={x}) then iso:=iso union {x};
fi;
od;
od;
iso;
end:
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<Text-field style="Heading 1" layout="Heading 1">Verification</Text-field>QyQqKEkiaUc2IiIiIkkpZGlzY3JldGVHRiVGJkkmc3BhY2VHRiVGJiEiIg==X:={a,b,c,d};S:={{a},{b},{c},{d}};B:=Basis(S);T:=Topology(B);IsTopology(T);A:={a,c};LimitPoints_A:=LimitPoints(A,X,T);ClosurePoints_A:=ClosurePoints(A,X,T);BoundaryPoints_A:=BoundaryPoints(A,X,T);InteriorPoints_A:=InteriorPoints(A,X,T);ExteriorPoints_A:=ExteriorPoints(A,X,T);IsolatedPoints_A:=IsolatedPoints(A,X,T); IsolatedPoints2_A:=IsolatedPoints2(A,X,T);NiI=X:={a,b,c,d};S:={{}};Basis(S);T:=Topology(B);IsTopology(T);A:={b};LimitPoints_A:=LimitPoints(A,X,T);ClosurePoints_A:=ClosurePoints(A,X,T);BoundaryPoints_A:=BoundaryPoints(A,X,T);InteriorPoints_A:=InteriorPoints(A,X,T);ExteriorPoints_A:=ExteriorPoints(A,X,T);IsolatedPoints_A:=IsolatedPoints(A,X,T); IsolatedPoints2_A:=IsolatedPoints2(A,X,T);NiI=X:={a,b};S:={{a}};B:=Basis(S);T:=Topology(T);IsTopology(T);A:={b};LimitPoints_A:=LimitPoints(A,X,T);ClosurePoints_A:=ClosurePoints(A,X,T);BoundaryPoints_A:=BoundaryPoints(A,X,T);InteriorPoints_A:=InteriorPoints(A,X,T);ExteriorPoints_A:=ExteriorPoints(A,X,T);IsolatedPoints_A:=IsolatedPoints(A,X,T);IsolatedPoints2_A:=IsolatedPoints2(A,X,T);NiI=X:={a,b,c,d};S:={{a},{a,c},{c,d}};B:=Basis(S);T:=Topology(B);IsTopology(T);A:={a,c,d};LimitPoints_A:=LimitPoints(A,X,T);ClosurePoints_A:=ClosurePoints(A,X,T);BoundaryPoints_A:=BoundaryPoints(A,X,T);InteriorPoints_A:=InteriorPoints(A,X,T);ExteriorPoints_A:=ExteriorPoints(A,X,T);IsolatedPoints_A:=IsolatedPoints(A,X,T);IsolatedPoints2_A:=IsolatedPoints2(A,X,T);JSFH