3. Prove that A x B=Ax B. Exercise 10 Hausdorff product topology Prove that the product of two Hausdorff spaces is Hausdorff. la.
3. Prove that A x B=Ax B. Exercise 10 Hausdorff product topology Prove that the product of two Hausdorff spaces is Hausdorff. la.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
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Chapter2: Second-order Linear Odes
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topology exercice 10
![Exercise 7. Basis and comparable topology| Let X = Rand K = {;n € N}. Consider
the following collections on X:
B = {]a, b[;a, b = R, a <b},
B' = {[a,b; a, b e R, 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 " 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 topological 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= A x 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 Haus dorff if and only if the diagonal A =
{(x,x); a X} is closed in X X X.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b5f7db1-a103-4627-9a7f-8775edb35b3d%2F5f8d838d-b8e5-4e5b-bfc8-628826940362%2F8v1g1id_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 7. Basis and comparable topology| Let X = Rand K = {;n € N}. Consider
the following collections on X:
B = {]a, b[;a, b = R, a <b},
B' = {[a,b; a, b e R, 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 " 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 topological 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= A x 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 Haus dorff if and only if the diagonal A =
{(x,x); a X} is closed in X X X.
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