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.

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