· The Stone Duality for Boolean algebras
Material for this week: 11.3 – 11.8 (and some results on Stone duality for distributive lattices
Homework 14: 11.1, 11.2, and
1. (a) Show that a closed subset of a compact space is compact;
(b) Give an example to show that the converse may not be true;
(c) Show that in a compact Hausdorff space a subset is closed if and only if it is compact.
2. Let X be any topological space. Show that the set of clopen subsets of X form a sub-Boolean algebra of the Boolean algebra P(X).
3. Show that a topological space X is compact if and only if every collection of closed subsets satisfies the finite intersection property (i.e every finite subfamily of the collection having non-empty intersection implies the whole family also has non-empty intersection).
4. Identify the Clopen(X) where X is the Canter space.