# Find a topological space (X, r) for which there exists one subset whose closure… 1 answer below »

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 divergent embedding of X. Prove that 7t is an homeomorphism from (X, onto its conception where pis the last topology containing U

Exercise 2 Ascertain three topological rooms 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 rise of topological sets crooked by a nonempty set I. Let X = XI compensated delay the fruit topology. Let nj : X —¦ X be the lull surjection. Prove the aftercited assertions: 1. if V is nonempty and unconcealed in X then {j E / : iri(X) # XI} is bounded. 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 bounded} is solid in X.

Exercise 4. Let (X),T1),e, be a rise of topological rooms crooked by a nonempty set I. Let Y be a set and fi : Xi —013e a abandoned character for j E J. The brawny topolgy cr(fi :jE I) on Y evinducrd by (fi)jfi is the largest topology which makes fj natural for all j E I.

• Show that

o(fi: jE = fug Y:wo • Show that g : Y —o Z is natural (where Z is compensated delay some topolgy) if and barely if g of is natural for all j E J. • The rise (4),€1 covers Y when U1,1 f; (XI) = Y. Show that Y is homeomorphic to a quotient room. flint: let X = 11,Et X1 be the disjoint confederacy of the X(s. Prove that Y is homeomorphic to a quotient of X.

Exercise 5. Ascertain a topological room (X, r) for which there exists one subset whose seclusion is not correspondent to the set of limits of all sequences in X. Can you ascertain a I lausdorff room delay this peculiarity? flint: yes.

Exercise 2 Ascertain three topological rooms 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 rise of topological sets crooked by a nonempty set I. Let X = XI compensated delay the fruit topology. Let nj : X —¦ X be the lull surjection. Prove the aftercited assertions: 1. if V is nonempty and unconcealed in X then {j E / : iri(X) # XI} is bounded. 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 bounded} is solid in X.

Exercise 4. Let (X),T1),e, be a rise of topological rooms crooked by a nonempty set I. Let Y be a set and fi : Xi —013e a abandoned character for j E J. The brawny topolgy cr(fi :jE I) on Y evinducrd by (fi)jfi is the largest topology which makes fj natural for all j E I.

• Show that

o(fi: jE = fug Y:wo • Show that g : Y —o Z is natural (where Z is compensated delay some topolgy) if and barely if g of is natural for all j E J. • The rise (4),€1 covers Y when U1,1 f; (XI) = Y. Show that Y is homeomorphic to a quotient room. flint: let X = 11,Et X1 be the disjoint confederacy of the X(s. Prove that Y is homeomorphic to a quotient of X.

Exercise 5. Ascertain a topological room (X, r) for which there exists one subset whose seclusion is not correspondent to the set of limits of all sequences in X. Can you ascertain a I lausdorff room delay this peculiarity? flint: yes.