集和拓朴概念

cjsh

1
拓朴空间：
集的子集族里的元素交并都在这子集族里，且{}，原集作为元素也在这子集族里
三元素集有19种，可只找出17种，哪位看看缺哪两？

S := { 1, 2,3};
S1:=Subsets(S);
S1;
t1:={{},{ 1 },{ 1, 3 }, { 1, 2, 3 }};
t1;
t2:={{},{ 2 },{ 1, 2 }, { 1, 2, 3 }};
t2;
t3:={{},{ 1 }, { 1, 2, 3 }};
t3;
t4:={{}, { 1, 2, 3 }};
t4;
t5:={{},{1}, { 1, 2, 3 }};
t5;
t6:={{},{1}, {2},{1,2},{ 1, 2, 3 }};
t6;
t7:={{},{1}, {3},{1,3},{ 1, 2, 3 }};
t7;
t8:={{},{2}, {3},{2,3},{ 1, 2, 3 }};
t8;
t9:={{},{1},{2}, {3},{1,2},{2,3},{1,3},{ 1, 2, 3 }};
t9;
t10:={{ 1 },{ 1, 3 }, { 1, 2, 3 }};
t10;
t11:={{ 2 },{ 1, 2 }, { 1, 2, 3 }};
t11;
t12:={{ 3 },{ 1, 3 }, { 1, 2, 3 }};
t12;
t13:={{ 1, 2, 3 }};
t13;
t14:={{1},{2}, {3},{1,2},{2,3},{1,3},{ 1, 2, 3 }};

t14;
t15:={{1}, {2},{1,2},{ 1, 2, 3 }};
t15;
t16:={{1}, {3},{1,3},{ 1, 2, 3 }};
t16;
t17:={{2}, {3},{2,3},{ 1, 2, 3 }};
t17;


{
    { 1 },
    {},
    { 1, 3 },
    { 2 },
    { 3 },
    { 1, 2, 3 },
    { 2, 3 },
    { 1, 2 }
}
{
    { 1 },
    {},
    { 1, 3 },
    { 1, 2, 3 }
}
{
    {},
    { 2 },
    { 1, 2, 3 },
    { 1, 2 }
}
{
    { 1 },
    {},
    { 1, 2, 3 }
}
{
    {},
    { 1, 2, 3 }
}
{
    { 1 },
    {},
    { 1, 2, 3 }
}
{
    { 1 },
    {},
    { 2 },
    { 1, 2, 3 },
    { 1, 2 }
}
{
    { 1 },
    {},
    { 1, 3 },
    { 3 },
    { 1, 2, 3 }
}
{
    {},
    { 2 },
    { 3 },
    { 1, 2, 3 },
    { 2, 3 }
}
{
    { 1 },
    {},
    { 1, 3 },
    { 2 },
    { 3 },
    { 1, 2, 3 },
    { 2, 3 },
    { 1, 2 }
}
{
    { 1 },
    { 1, 3 },
    { 1, 2, 3 }
}
{
    { 2 },
    { 1, 2, 3 },
    { 1, 2 }
}
{
    { 1, 3 },
    { 3 },
    { 1, 2, 3 }
}
{
    { 1, 2, 3 }
}
{
    { 1 },
    { 1, 3 },
    { 2 },
    { 3 },
    { 1, 2, 3 },
    { 2, 3 },
    { 1, 2 }
}
{
    { 1 },
    { 2 },
    { 1, 2, 3 },
    { 1, 2 }
}
{
    { 1 },
    { 1, 3 },
    { 3 },
    { 1, 2, 3 }
}
{
    { 2 },
    { 3 },
    { 1, 2, 3 },
    { 2, 3 }
}

cjsh

A topological space, also called an abstract topological space, is a set  together with a collection of open subsets  that satisfies the four conditions:
1. The empty set  is in .
2.  is in .
3. The intersection of a finite number of sets in  is also in .
4. The union of an arbitrary number of sets in  is also in .
Alternatively,  may be defined to be the closed sets rather than the open sets, in which case conditions 3 and 4 become:
3. The intersection of an arbitrary number of sets in  is also in .
4. The union of a finite number of sets in  is also in .