restart;

with(combinat):

 #(1)A procedure to generate a topology from a given subbasis S.

generaTe:=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:

#(2)A procedure to check if a given collection over X is Topology or not .

CheckTopology:=proc(T)

local i,O,CT,CuT;

CT:={};

if not`subset`(T,powerset(X)) then false; else

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 union {{},X};

evalb(CT=T);

fi;

end:

#A procedure to finds the clopen sets of the topology(T)

CO:=proc(X,T)

local A,W;W:={};

for A in T do

if member(X minus A,T)then W:=W union{A};fi;

od;

W;

end:

 #A procedure to obtain the relative topology on a subset of X  .

 subspace:=proc(A,X,T)

map2(`intersect`,A,T);

end:

#A procedure to chek that if the topology is connected [1].

 isConn:=proc(X,T)

evalb(CO(X,T)={X,{}});

end:

#A procedure to find the connected components of a given point .

 K:=proc(x,X,T)
local i,S,SK;
SK:={};
S:=map2(`union`,{x},powerset(X));
for i to nops(S) do
if isConn(S[i],subspace(S[i],X,T)) then SK:=SK union S[i];fi;
od; SK ;
end:

#(i)A procedure to list all topologies on a finite set.

 ALLT:=proc(probableT)
local T,ALLTOPO;

ALLTOPO:={};

for T in probableT do

if CheckTopology(T)=true then ALLTOPO:=ALLTOPO union {T};else

ALLTOPO:=ALLTOPO;

fi;

od;

ALLTOPO;

end:

 #(ii)A procedure to list the connected topologies on a finite set X .

 AllConnected:=proc( AllTopologies)

local B,T;

B:={};

for T in AllTopologies do

if CO(X,T)={X,{}} then B:= B union {T}; else B:=B;

fi;

od;

B;

end:


#(iii)A procedure to find the connected components of a given space.

ALLCC:=proc(X,T)
local x,CC;
CC:={};
for x in X do
CC:=CC union {K(x,X,T)};
od;
CC;
end:

X:={a};
 

 Y:=powerset(X):

 Z:=Y minus{{},X}:

 W:=powerset(Z):

 probableT:={seq(w union{{},X},w=W)}:

 nops(probableT):

AllTopologies:=ALLT(probableT);

print(`the number of topologies is`,nops(AllTopologies),`over a set with `,nops(X),`points`);

Allconnectedtopologies:=AllConnected(AllTopologies);

print(`there are `,nops(Allconnectedtopologies),`connected spaces on a set X with `,nops(X),`points` ):

X:={a,b};
 

Y:=powerset(X):
 

 Z:=Y minus{{},X}:

 W:=powerset(Z):

 probableT:={seq(w union{{},X},w=W)}:
 

 nops(probableT):

AllTopologies:=ALLT(probableT);

print(`the number of topologies is`,nops(AllTopologies),`over a set with `,nops(X),`points`);

Allconnectedtopologies:=AllConnected(AllTopologies);

print(`there are `,nops(Allconnectedtopologies),`connected spaces on a set X with `,nops(X),`points` );

X:={a,b,c};

Y:=powerset(X):

Z:=Y minus{{},X}:

 W:=powerset(Z):

probableT:={seq(w union{{},X},w=W)}:

nops(probableT):

AllTopologies:=ALLT(probableT);

print(`the number of topologies is`,nops(AllTopologies),`over a set with `,nops(X),`points`);

Allconnectedtopologies:=AllConnected(AllTopologies);

print(`there are `,nops(Allconnectedtopologies),`connected spaces on a set X with `,nops(X),`points` );

#All connected components;

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

T:={{},X};

ALLConnected_Components:=ALLCC(X,T);

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

T:=powerset(X);

ALLConnected_Components:=ALLCC(X,T);

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

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

ALLConnected_Components:=ALLCC(X,T);