1. If (X, T ) and (Y, U ) are topological spaces, A is a base for T and B is a base for U , show that the collection of all sets A × B for A ∈ A and B ∈ B is a base for the product topology on X × Y .
2. (a) Prove that any intersection of topologies on a set is a topology.
(b) Prove that for any collection U of subsets of a set X , there is a smallest topology on X including U , using part (a) (rather than subbases).
3. (a) Let An be the set of all integers greater than n. Let Bn be the collection of all subsets of {1, ..., n}. Let Tn be the collection of sets of positive integers that are either in Bn or of the form An ∪ B for some B ∈ Bn . Prove that Tn is a topology.
(b) Show that Tn for n = 1, 2,... , is an inclusion-chain of topologies whose union is not a topology.
(c) Describe the smallest topology which includes Tn for all n.