restart;
with(combinat):

#(1) A procedure to generate Excluded point topology.
GeneratExcT:=proc(n,m)
local i,x,X,EXT,EXP;
if m<=n then
X:={seq(x[i],i=1..n)};
EXP:=x[m];
print(`The excluded point is`,x[m],`.And the excluded point topology is:`);
EXT:=powerset(X minus {EXP}) union {X};
else print(`false entries`);
fi;
end:

#########################~~~~~~##################################################################################;

#(2) A procedure to Check if a given topology is Excluded point topology or not .
CheckExcluded:=proc(X,T)
local x,C;
C:={};
for x in X do
if T=combinat:-powerset(X minus {x}) union {X} then C:=C union {x};else C:=C;
end if;
od;
if nops(C)=1 then  print(`True The given topology is excluded  point topology and the excluded point is `,op(C)); else false;
end if;
end:

########################~~~~~~~~~##########################~~~~~~~~~~############################################;

#(3) A procedure to find the number of proper open sets in a given excluded point topology.

NumberofPOinExcluded:=proc(X,EXT)
print(`The number of proper open sets in the given excluded  point topology is`,nops(EXT)- 2,`over a set with`,nops(X),`points`);
end:

########################~~~~~~~~~##########################~~~~~~~~~~############################################;

#(4) A procedure to find all excluded point topologies over a given  set.

AllExcludedPointTopologies:=proc(X)

local x,CountEx;
CountEx:={};
for x in X do
CountEx:={powerset(X minus {x}) union {X} } union CountEx;
od;
CountEx;
end:

##########~~~~~~~#############################~~~~~~~##################################~~~~~~#########;

#(5) A procedure to generate included point topology.
GeneratIncT:=proc(n,m)
local cc,i,T1,XX;
if m<=n then
XX:={seq(x[i],i=1..n)};
cc:={x[m]};
print(`The included point is`,x[m],`.And the included point topology is:`);
T1:=map(`union`,powerset(XX),cc) union {{}};
else print(`false entries`);
T1;XX;
fi;
end:
########################~~~~~~~~~##########################~~~~~~~~~~############################################;

#(6) A procedure to check if a given topology is included point topology or not.
CheckIncluded:=proc(X,T)
local x,C;C:={};
for x in X do
if T=map(`union`,combinat:-powerset(X),{x}) union {{}} then C:=C union {x} ;else C:=C;
end if;
od;
C;
if nops(C)=1 then print(`True The given topology is included point topology and the included point is`,op(C));else false;
end if;
end:

#(7) A procedure to find the number of proper open sets in  a given included point topology.

NumberofPOinIncluded:=proc(X,T)
print(`The number of proper open sets in the given included  point topology is`,nops(T)- 2,`over a set with`,nops(X),`points`);
end:

#(8) A procedure to find all included point topologies over a given set.

AllincludedPointTopologies:=proc(X)

local x,Countinc;
Countinc:={};
for x in X do
Countinc:={map(`union`,powerset(X),{x}) union {{}}} union Countinc;
od;
Countinc;
end:

GeneratExcT(7,2);

X:={a,b,c,d};

T:={{},{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c},X};

CheckExcluded(X,T);

NumberofPOinExcluded(X,T);

AllExcludedPointTopologies(X);

###############################################################################################################.
GeneratIncT(5,3);

X:={a,b,c,d,e};

T:={{},{a},{a,b},{a,c},{a,d},{a,e},{a,b,c},{a,b,d},{a,b,e},{a,d,e},{a,c,e},{a,c,d},{a,b,c,d},{a,b,c,e},{a,c,d,e},{a,b,d,e},X};

CheckIncluded(X,T);

NumberofPOinIncluded(X,T);
AllincludedPointTopologies(X);

#
Y:={1,2,3}; TY:=powerset(Y);

CheckExcluded(Y,TY);

CheckIncluded(Y,TY);

GeneratIncT(6,6);

GeneratIncT(1,1);

GeneratIncT(5,5);

GeneratExcT(1,1);

GeneratExcT(3,6);