WEEK 51:

 

·         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.