= Exercise 12. |Diagonal Show that X is Hausdorff if and only if the diagonal A {(2,1); z X) is closed in X X X.
= Exercise 12. |Diagonal Show that X is Hausdorff if and only if the diagonal A {(2,1); z X) is closed in X X X.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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topology exercice 12
![**Exercise 7.** *[Basis and comparable topology]* Let \( X = \mathbb{R} \) and \( K = \left\{ \frac{1}{n}; n \in \mathbb{N} \right\} \). Consider the following collections on \( X \):
\[
\mathcal{B} = \{ [a, b) ; a, b \in \mathbb{R}, a < b \},
\]
\[
\mathcal{B}' = \{ [a, b]; a, b \in \mathbb{R}, a < b \},
\]
and
\[
\mathcal{B}'' = \{ [a, b]; a, b \in \mathbb{R}, a < b \} \cup \{ [a, b] \setminus K ; a, b \in \mathbb{R}, a < b \}.
\]
Knowing that \( \mathcal{B} \) and \( \mathcal{B}' \) are bases for some topology on \( X \), prove that \( \mathcal{B}'' \) is a basis for a topology on \( X \). Furthermore, let \( \tau, \tau', \) and \( \tau'' \) denote the topologies on \( X \) generated by \( \mathcal{B}, \mathcal{B}', \) and \( \mathcal{B}'' \), respectively. Prove that \( \tau'' \) and \( \tau'' \) are finer than \( \tau \), and that \( \tau' \) and \( \tau'' \) are not comparable.
**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 \( A \times B \) is the same as the topology \( A \times B \) inherits as a subspace of \( X \times Y \).
**Exercise 9.** *[Closed product topology]* Let \( X \) and \( Y \) be topological space, \( A, U \) be subsets of \( X \), and \( B \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b5f7db1-a103-4627-9a7f-8775edb35b3d%2F9efd1eb6-0f04-4995-93ef-b4a507e5f80b%2Fb10jx2d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 7.** *[Basis and comparable topology]* Let \( X = \mathbb{R} \) and \( K = \left\{ \frac{1}{n}; n \in \mathbb{N} \right\} \). Consider the following collections on \( X \):
\[
\mathcal{B} = \{ [a, b) ; a, b \in \mathbb{R}, a < b \},
\]
\[
\mathcal{B}' = \{ [a, b]; a, b \in \mathbb{R}, a < b \},
\]
and
\[
\mathcal{B}'' = \{ [a, b]; a, b \in \mathbb{R}, a < b \} \cup \{ [a, b] \setminus K ; a, b \in \mathbb{R}, a < b \}.
\]
Knowing that \( \mathcal{B} \) and \( \mathcal{B}' \) are bases for some topology on \( X \), prove that \( \mathcal{B}'' \) is a basis for a topology on \( X \). Furthermore, let \( \tau, \tau', \) and \( \tau'' \) denote the topologies on \( X \) generated by \( \mathcal{B}, \mathcal{B}', \) and \( \mathcal{B}'' \), respectively. Prove that \( \tau'' \) and \( \tau'' \) are finer than \( \tau \), and that \( \tau' \) and \( \tau'' \) are not comparable.
**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 \( A \times B \) is the same as the topology \( A \times B \) inherits as a subspace of \( X \times Y \).
**Exercise 9.** *[Closed product topology]* Let \( X \) and \( Y \) be topological space, \( A, U \) be subsets of \( X \), and \( B \)
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