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Exercise 11. Generalization: Hausdorff product topology] Let (X.)ier be an arbitrary family of Hausdorff spaces. Prove that IIX, is Hausdorff space for the product topology. iel

Exercise 11. Generalization: Hausdorff product topology] Let (X.)ier be an arbitrary family of Hausdorff spaces. Prove that IIX, is Hausdorff space for the product topology. iel

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.4: Complex Numbers
Problem 60E
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topology exercice 11

Exercise 7. Basis and comparable topology| Let X = Rand K = {1; n € N}. Consider the following collections on X: B = {]a, b[; a, b = R, a <b}, B' = {[a, b; a, b = R₁ a <b}, B" = {]a, b[; a, b € R, a <b} U {]a,b[\K; a, b € R, a <b}. Knowing that B and B' are bases for some topology on X, prove that B" is basis for a topology on X. Furthermore, let 7, 7', and 7" denote the topologies on X generated by B. B', and B", respectively. Prove that and 7" are finer than 7, and that 7 and 7" are not comparable. and Exercise 8 Subspaces product topology Let A be a subspace of X and let B be a subspace of Y. We equip A and B with the subspace topologies. Prove that the product topology on A x B is the same as the topology A x B inherits as a subspace of X x Y. Exercise 9 Closed product topology| Let X and Y be topologic al space, A, U be subsets of X, and B be a subset of Y. 1. Show that if A is closed in X and B is closed in Y then A x B is closed in X x Y. 2. Show that if U is open in X and A is closed in X, then U\A is open in X, and A\U is closed in X. 3. Prove that Ax B= Āx B. Exercise 10 Hausdorff product topology Prove that the product of two Hausdorff spaces is Hausdorff. Exercise 11 Generalization: Hausdorff product topology Let (X)ier be an arbitrary family of Hausdorff spaces. Prove that IX, is Hausdorff space for the product topology. iel Exercise 12 |Diagonal Show that X is Hausdorff if and only if the diagonal A = {(x,x); z e X} is closed in XX X.
Transcribed Image Text:Exercise 7. Basis and comparable topology| Let X = Rand K = {1; n € N}. Consider the following collections on X: B = {]a, b[; a, b = R, a <b}, B' = {[a, b; a, b = R₁ a <b}, B" = {]a, b[; a, b € R, a <b} U {]a,b[\K; a, b € R, a <b}. Knowing that B and B' are bases for some topology on X, prove that B" is basis for a topology on X. Furthermore, let 7, 7', and 7" denote the topologies on X generated by B. B', and B", respectively. Prove that and 7" are finer than 7, and that 7 and 7" are not comparable. and Exercise 8 Subspaces product topology Let A be a subspace of X and let B be a subspace of Y. We equip A and B with the subspace topologies. Prove that the product topology on A x B is the same as the topology A x B inherits as a subspace of X x Y. Exercise 9 Closed product topology| Let X and Y be topologic al space, A, U be subsets of X, and B be a subset of Y. 1. Show that if A is closed in X and B is closed in Y then A x B is closed in X x Y. 2. Show that if U is open in X and A is closed in X, then U\A is open in X, and A\U is closed in X. 3. Prove that Ax B= Āx B. Exercise 10 Hausdorff product topology Prove that the product of two Hausdorff spaces is Hausdorff. Exercise 11 Generalization: Hausdorff product topology Let (X)ier be an arbitrary family of Hausdorff spaces. Prove that IX, is Hausdorff space for the product topology. iel Exercise 12 |Diagonal Show that X is Hausdorff if and only if the diagonal A = {(x,x); z e X} is closed in XX X.
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