JSFH

Basic Procedures

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)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

CO:=proc(X,T)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

local A,W;W:={};

for A in T do

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

od;

end:

#A procedure to obtain the relative topology on a subset of X LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGNEY0LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRjdGNA==.

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiUtSSNtbkdGJDYkUSIxRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGNEY0LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRjdGNA==.

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: 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

#(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;

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:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2KFEvRGVtb25zdHJhdGlvbnNGJy8lJXNpemVHUSMxNEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGQi8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GPjYmRkBGQ0ZGL0ZJUSVhdXRvRictRiw2JVEhRicvRjZGNC9GOVEnaXRhbGljRicvJStleGVjdXRhYmxlR0Y3L0Y5USdub3JtYWxGJw==

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

X:={a};

Y:=powerset(X):

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

W:=powerset(Z):

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

nops(probableT):

SS9EZW1vbnN0cmF0aW9uc0c2Ig==

PCNJImFHNiI=

AllTopologies:=ALLT(probableT);

PCM8JDwiPCNJImFHNiI=

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

NidJPHRoZX5udW1iZXJ+b2Z+dG9wb2xvZ2llc35pc0c2IiIiIkkxb3Zlcn5hfnNldH53aXRofkdGJEYlSSdwb2ludHNHRiQ=

Allconnectedtopologies:=AllConnected(AllTopologies);

PCM8JDwiPCNJImFHNiI=

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

NidJK3RoZXJlfmFyZX5HNiIiIiJJQmNvbm5lY3RlZH5zcGFjZXN+b25+YX5zZXR+WH53aXRofkdGJEYlSSdwb2ludHNHRiQ=

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

X:={a,b};

PCRJImFHNiJJImJHRiQ=

Y:=powerset(X):

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

W:=powerset(Z):

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

nops(probableT):

AllTopologies:=ALLT(probableT);

PCY8JDwiPCRJImFHNiJJImJHRic8JUYkPCNGJkYlPCVGJDwjRihGJTwmRiRGKkYsRiU=

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

NidJPHRoZX5udW1iZXJ+b2Z+dG9wb2xvZ2llc35pc0c2IiIiJUkxb3Zlcn5hfnNldH53aXRofkdGJCIiI0kncG9pbnRzR0Yk

Allconnectedtopologies:=AllConnected(AllTopologies);

PCU8JDwiPCRJImFHNiJJImJHRic8JUYkPCNGJkYlPCVGJDwjRihGJQ==

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

NidJK3RoZXJlfmFyZX5HNiIiIiRJQmNvbm5lY3RlZH5zcGFjZXN+b25+YX5zZXR+WH53aXRofkdGJCIiI0kncG9pbnRzR0Yk

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiI0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2J1EtRXhhbXBsZSgzKX46RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0Y0L0YwUSVib2xkRicvJStmb250d2VpZ2h0R0ZQLyUrZXhlY3V0YWJsZUdGNEYv

X:={a,b,c};

PCVJImFHNiJJImJHRiRJImNHRiQ=

Y:=powerset(X):

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

W:=powerset(Z):

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

nops(probableT):

AllTopologies:=ALLT(probableT);

PD88JDwiPCVJImFHNiJJImJHRidJImNHRic8JUYkPCNGJkYlPCVGJDwjRihGJTwlRiQ8I0YpRiU8JUYkPCRGJkYoRiU8JUYkPCRGJkYpRiU8JUYkPCRGKEYpRiU8JkYkRitGMUYlPCZGJEYrRjNGJTwmRiRGK0Y1RiU8JkYkRi1GMUYlPCZGJEYtRjNGJTwmRiRGLUY1RiU8JkYkRi9GMUYlPCZGJEYvRjNGJTwmRiRGL0Y1RiU8J0YkRitGLUYxRiU8J0YkRitGL0YzRiU8J0YkRitGMUYzRiU8J0YkRi1GL0Y1RiU8J0YkRi1GMUY1RiU8J0YkRi9GM0Y1RiU8KEYkRitGLUYxRjNGJTwoRiRGK0YtRjFGNUYlPChGJEYrRi9GMUYzRiU8KEYkRitGL0YzRjVGJTwoRiRGLUYvRjFGNUYlPChGJEYtRi9GM0Y1RiU8KkYkRitGLUYvRjFGM0Y1RiU=

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

NidJPHRoZX5udW1iZXJ+b2Z+dG9wb2xvZ2llc35pc0c2IiIjSEkxb3Zlcn5hfnNldH53aXRofkdGJCIiJEkncG9pbnRzR0Yk

Allconnectedtopologies:=AllConnected(AllTopologies);

PDU8JDwiPCVJImFHNiJJImJHRidJImNHRic8JUYkPCNGJkYlPCVGJDwjRihGJTwlRiQ8I0YpRiU8JUYkPCRGJkYoRiU8JUYkPCRGJkYpRiU8JUYkPCRGKEYpRiU8JkYkRitGMUYlPCZGJEYrRjNGJTwmRiRGLUYxRiU8JkYkRi1GNUYlPCZGJEYvRjNGJTwmRiRGL0Y1RiU8J0YkRitGLUYxRiU8J0YkRitGL0YzRiU8J0YkRitGMUYzRiU8J0YkRi1GL0Y1RiU8J0YkRi1GMUY1RiU8J0YkRi9GM0Y1RiU=

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

NidJK3RoZXJlfmFyZX5HNiIiIz5JQmNvbm5lY3RlZH5zcGFjZXN+b25+YX5zZXR+WH53aXRofkdGJCIiJEkncG9pbnRzR0Yk

#All connected components;

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

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

PCZJImFHNiJJImJHRiRJImNHRiRJImRHRiQ=

T:={{},X};

PCQ8IjwmSSJhRzYiSSJiR0YmSSJjR0YmSSJkR0Ym

ALLConnected_Components:=ALLCC(X,T);

PCM8JkkiYUc2IkkiYkc2IkkiY0c2IkkiZEc2Ig==

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

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

PCZJImFHNiJJImJHRiRJImNHRiRJImRHRiQ=

T:=powerset(X);

PDI8IjwjSSJhRzYiPCNJImJHRiY8I0kiY0dGJjwjSSJkR0YmPCRGJUYoPCRGJUYqPCRGJUYsPCRGKEYqPCRGKEYsPCRGKkYsPCVGJUYoRio8JUYlRihGLDwlRiVGKkYsPCVGKEYqRiw8JkYlRihGKkYs

ALLConnected_Components:=ALLCC(X,T);

PCY8I0kiYUc2IjwjSSJiR0YlPCNJImNHRiU8I0kiZEdGJQ==

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

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

PCZJImFHNiJJImJHRiRJImNHRiRJImRHRiQ=

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

PCg8IjwjSSJhRzYiPCRGJUkiYkdGJjwkSSJjR0YmSSJkR0YmPCVGJUYqRis8JkYlRihGKkYr

ALLConnected_Components:=ALLCC(X,T);

PCQ8JEkiYUc2IkkiYkdGJTwkSSJjR0YlSSJkR0Yl

JSFH