Topology: Compact Set, Open Covering, Lindelöf Theorem, Lindelöf Space, Finite Intersection Property

- Let F be a bounded and closed subset of R. Let {B_a : a in A} be an open covering of F. Show that there sre finite elements B_i in {B_a : a in A} such that {B_1, ..., B_n} is an open covering of F.
- Let A be a subset of R. Let {B_a : a in A} be an open covering of A. Show that there are at most countable elements B_i in {B_a : a in A} such that {B_i : i = 1, 2, ...} is an open covering of A.
- Let X be a topological space. A collection C of subsets of X is said to have the finite intersection property if for every finite subcollection {C-1, ..., C_n} of C, the intersection C_1 \cap ... \cap C_n is nonempty.
Let {F_a : a in A} be a collection of subsets of R with bounded and closed sets F_a for all a in A. Suppose that the collection {F_a : a in A} has the finite intersection property. Show that \cap_{a in A} F_a is not empty.