(a) Given a topology on , a collection is called a basis for if every non-empty set in is a union of sets in . Prove that a collection is a basis for some topology if it satisfies:
(i) the union of all sets in is ;
(ii) if for two sets and in , then there is a set with .
(b) On consider the dictionary order given by
if or if and . Given points and in let
Show that the sets for and in form a basis of a topology.
(c) Show that this topology on does not have a countable basis.