restart;

with(combinat):




#(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:

#(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:

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);

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);

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);

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);