Exercise 1. Let X be a set. Let .9' be a set of topologies on X and let r = Opeyp. Let rr : x E X E flers X be the diagonal embedding of X. Prove that 7t is an homeomorphism from (X, onto its image where pis the smallest topology containing U
Exercise 2 Find three topological spaces X, Y, Z such that X x Y is homeomorphic to Z x Y yet X and Z are not homeomorphic.
Exercise 3. Let (X„ be some family of topological sets indexed by a nonempty set I. Let X = XI endowed with the product topology. Let nj : X —¦ X be the canonical surjection. Prove the following assertions: 1. if V is nonempty and open in X then {j E / : iri(X) # XI} is finite. 2. Let x E X and let jo E J. Prove that the set {y E X : Vj E J j0 jo x, — y } is homeomorphic to X. 3. Let x E X. Prove that the set (y E X: { jE I:xj qE yi} is finite} is dense in X.
Exercise 4. Let (X),T1),e, be a family of topological spaces indexed by a nonempty set I. Let Y be a set and fi : Xi —013e a given function for j E J. The strong topolgy cr(fi :jE I) on Y evinducrd by (fi)jfi is the largest topology which makes fj continuous for all j E I.
• Show that
o(fi: jE = fug Y:wo • Show that g : Y —o Z is continuous (where Z is endowed with some topolgy) if and only if g of is continuous for all j E J. • The family (4),€1 covers Y when U1,1 f; (XI) = Y. Show that Y is homeomorphic to a quotient space. flint: let X = 11,Et X1 be the disjoint union of the X(s. Prove that Y is homeomorphic to a quotient of X.
Exercise 5. Find a topological space (X, r) for which there exists one subset whose closure is not equal to the set of limits of all sequences in X. Can you find a I lausdorff space with this property? flint: yes.