conditions for a collection of subsets to be a basis for some topology
Not just any collection of subsets of can be a basis for a topology on . For instance, if we took to be all open intervals of length in , isn’t the basis for any topology on : and are unions of elements of , but their intersection is not. The collection formed by arbitrary unions of members of isn’t closed under finite intersections and isn’t a topology.
We’d like to know which collections of subsets of could be the basis for some topology on . Here’s the result:
First, we’ll show that if is the basis for some topology on , then it satisfies the two conditions listed.
is a topology on , so . Since is a basis for , that means can be written as a union of members of : since every is in this union, every is contained in some member of . That takes care of the first condition.
For the second condition: if and are elements of , they’re also in . is closed under intersection, so is open in . Then can be written as a union of members of , and any is contained by some basis element in this union.
Second, we’ll show that if a collection of subsets of satisfies the two conditions, then the collection of unions of members of is a topology on .
: is the null union of zero elements of .
: by the first condition, every is contained in some member of . The union of all the members of is then all of .
is closed under arbitrary unions: Say we have a union of sets …
(since each is a union of sets in )
Since that’s a union of elements of , it’s also a member of .
is closed under finite intersections: since a collection of sets is closed under finite intersections if and only if it is closed under pairwise intersections, we need only check that the intersection of two members of is in .
Any is contained in some and . By the second condition, gets us a with . Then
which is in .
|Title||conditions for a collection of subsets to be a basis for some topology|
|Date of creation||2013-03-22 14:21:49|
|Last modified on||2013-03-22 14:21:49|
|Last modified by||waj (4416)|