Theorem 3.30. Let (X,J) be a topological space, and let (Y, Fy) be a subspace. If B is a basis for T', then By = {Bn Y|B E B} is a basis for Ty.
Theorem 3.30. Let (X,J) be a topological space, and let (Y, Fy) be a subspace. If B is a basis for T', then By = {Bn Y|B E B} is a basis for Ty.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Could you show me how to do 3.30 in detail?
![**Theorem 3.30.** Let \((X, \mathscr{T})\) be a topological space, and let \((Y, \mathscr{T}_Y)\) be a subspace. If \(\mathscr{B}\) is a basis for \(\mathscr{T}\), then \(\mathscr{B}_Y = \{B \cap Y \mid B \in \mathscr{B}\}\) is a basis for \(\mathscr{T}_Y\).
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**Definition.** Let \((X, \mathscr{T})\) be a topological space. For \(Y \subseteq X\), the collection
\[
\mathscr{T}_Y = \{U \mid U = V \cap Y \text{ for some } V \in \mathscr{T}\}
\]
is a topology on \(Y\) called the **subspace topology**. It is also called the **relative topology** on \(Y\) inherited from \(X\). The space \((Y, \mathscr{T}_Y)\) is called a (topological) **subspace** of \(X\). If \(U \in \mathscr{T}_Y\) we say \(U\) is **open in \(Y\)**.
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**Theorem 3.25.** Let \((X, \mathscr{T})\) be a topological space and \(Y \subseteq X\). Then the collection of sets \(\mathscr{T}_Y\) is in fact a topology on \(Y\).
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**Theorem 3.28.** Let \((Y, \mathscr{T}_Y)\) be a subspace of \((X, \mathscr{T})\). A subset \(C \subseteq Y\) is closed in \((Y, \mathscr{T}_Y)\) if and only if there is a set \(D \subseteq X\), closed in \((X, \mathscr{T})\), such that \(C = D \cap Y\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfa77723-e387-48dc-857e-67a9efe99fb1%2Fa32a0384-8c94-4887-8ff4-82e06104de64%2F7vtp61_processed.png&w=3840&q=75)
Transcribed Image Text:**Theorem 3.30.** Let \((X, \mathscr{T})\) be a topological space, and let \((Y, \mathscr{T}_Y)\) be a subspace. If \(\mathscr{B}\) is a basis for \(\mathscr{T}\), then \(\mathscr{B}_Y = \{B \cap Y \mid B \in \mathscr{B}\}\) is a basis for \(\mathscr{T}_Y\).
---
**Definition.** Let \((X, \mathscr{T})\) be a topological space. For \(Y \subseteq X\), the collection
\[
\mathscr{T}_Y = \{U \mid U = V \cap Y \text{ for some } V \in \mathscr{T}\}
\]
is a topology on \(Y\) called the **subspace topology**. It is also called the **relative topology** on \(Y\) inherited from \(X\). The space \((Y, \mathscr{T}_Y)\) is called a (topological) **subspace** of \(X\). If \(U \in \mathscr{T}_Y\) we say \(U\) is **open in \(Y\)**.
---
**Theorem 3.25.** Let \((X, \mathscr{T})\) be a topological space and \(Y \subseteq X\). Then the collection of sets \(\mathscr{T}_Y\) is in fact a topology on \(Y\).
---
**Theorem 3.28.** Let \((Y, \mathscr{T}_Y)\) be a subspace of \((X, \mathscr{T})\). A subset \(C \subseteq Y\) is closed in \((Y, \mathscr{T}_Y)\) if and only if there is a set \(D \subseteq X\), closed in \((X, \mathscr{T})\), such that \(C = D \cap Y\).
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