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Question1-:(Filter generated by a family of sets). Let S ⊆ ℘(X) be nonempty family of subsets of X, and let BS be the family of all finite intersections of elements of S (note that S ⊆ BS). Show that BS is a basis for a filter. Show that this filter is nontrivial if and only if S has the finite intersection property.1 Moreover, prove that this is the smallest filter containing S. (Note that therefore there exist families that are not contained in any nontrivial filter).  Question2-:Let F be a nontrivial on X. Prove that the follow-ing statements are equivalent1. F is an ultrafilter.2. (∀A⊆X)A∈F ↔ X−A∈/F.3. (∀A,B⊆X)A∪B∈F → A∈F or B∈F.

Question

Question1-:

(Filter generated by a family of sets). Let S ⊆ ℘(X) be nonempty family of subsets of X, and let BS be the family of all finite intersections of elements of S (note that S ⊆ BS). Show that BS is a basis for a filter. Show that this filter is nontrivial if and only if S has the finite intersection property.1 Moreover, prove that this is the smallest filter containing S. (Note that therefore there exist families that are not contained in any nontrivial filter).

 

 

Question2-:

Let F be a nontrivial on X. Prove that the follow-

ing statements are equivalent
1. F is an ultrafilter.
2. (∀A⊆X)A∈F ↔ X−A∈/F.
3. (∀A,B⊆X)A∪B∈F → A∈F or B∈F.

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Step 1

As per norms, question 1 ( 3 subparts ) are answered. The problem concerns filters generated by families of subsets of a set.

Step 2

when is a  collection F of subsets of a set X  called a filter on X?

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Step 3

Criterion for a family of subsets B to form a base for a filter F on X. Basi...

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