Finite Excluded and Included Point Topologies
Taha Guma el turkiDepartment of Mathematics, Faculty of Science, University of Benghazi, LibyaE-mail: Taha1978_2002@yahoo.comIn this application we will compute new issues related to Finite Topological Spaces. The new procedures are :
(1) A procedure to generate Excluded point topology. "GeneratExcT(n,m)".
(2) A procedure to Check if a given topology is Excluded point topology or not "CheckExcluded(X,T)".
(3) A procedure to find the number of proper open sets in a given excluded point topology. "NumberofPOinExcluded(X,T)".
(4) A procedure to find all excluded point topologies over a given set. "AllExcludedPointTopologies(X)".
(5) A procedure to generate included point topology. "GeneratIncT(n,m)".
(6) A procedure to check if a given topology is included point topology or not. "CheckIncluded(X,T)".
(7) A procedure to find the number of proper open sets in a given included point topology. "NumberofPOinIncluded(X,T)".
(8) A procedure to find all included point topologies over a given set. "AllincludedPointTopologies(X,T)".
Notes: In procedures (1) and (5) the arguments n and m refer to the number of the set elements and "the order of the point which we want to exclude or include" respectively and these two numbers must satisfy that m<=n .KeyWords: Excluded point topology, Included point topology
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:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2L1EiI0YnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZILUkjbWlHRiQ2J1EsRXhhbXBsZSgxKTpGJ0YvLyUnaXRhbGljR0YxL0YzUSxib2xkLWl0YWxpY0YnRjUtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGSC8lJmRlcHRoR0ZYLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZUNiZGVkZZRmVuL0ZoblElYXV0b0YnRlMtRkw2I1EhRicvJStleGVjdXRhYmxlR0Y5L0YzUSdub3JtYWxGJw==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);#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
Y:={1,2,3}; TY:=powerset(Y);CheckExcluded(Y,TY);CheckIncluded(Y,TY);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2L1EiI0YnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZILUkjbWlHRiQ2J1EsRXhhbXBsZSgzKTpGJ0YvLyUnaXRhbGljR0YxL0YzUSxib2xkLWl0YWxpY0YnRjUvJStleGVjdXRhYmxlR0Y5L0YzUSdub3JtYWxGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=GeneratIncT(6,6);GeneratIncT(1,1);GeneratIncT(5,5);GeneratExcT(1,1);GeneratExcT(3,6);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=