Definition. Let (X,F) be a topological space. For Y c X, the collection Ty = {U|U = VoY for some V E I} %3D is a topology on Y called the subspace topology. It is also called the relative topology on Y inherited from X. The space (Y, Fy) is called a (topological) subspace of X. If U E Ty we say U is open in Y. Theorem 3.25. Let (X,J) be a topological space and Y c X. Then the collection of sets Ty is in fact a topology on Y.
Definition. Let (X,F) be a topological space. For Y c X, the collection Ty = {U|U = VoY for some V E I} %3D is a topology on Y called the subspace topology. It is also called the relative topology on Y inherited from X. The space (Y, Fy) is called a (topological) subspace of X. If U E Ty we say U is open in Y. Theorem 3.25. Let (X,J) be a topological space and Y c X. Then the collection of sets Ty is in fact a topology on Y.
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 3.25 in detail?
![**Definition.** Let \((X, \mathcal{T})\) be a topological space. For \(Y \subseteq X\), the collection
\[
\mathcal{T}_Y = \{U \mid U = V \cap Y \text{ for some } V \in \mathcal{T}\}
\]
is a topology on \(Y\) called the **subspace topology**. It is also called the **relative topology** on \(Y\) inherited from \(X\). The space \((Y, \mathcal{T}_Y)\) is called a (topological) **subspace** of \(X\). If \(U \in \mathcal{T}_Y\) we say \(U\) is **open in \(Y\)**.
**Theorem 3.25.** Let \((X, \mathcal{T})\) be a topological space and \(Y \subseteq X\). Then the collection of sets \(\mathcal{T}_Y\) is in fact a topology on \(Y\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfa77723-e387-48dc-857e-67a9efe99fb1%2F0e26fce5-e255-4587-b169-56db01556f01%2Fn79kna4_processed.png&w=3840&q=75)
Transcribed Image Text:**Definition.** Let \((X, \mathcal{T})\) be a topological space. For \(Y \subseteq X\), the collection
\[
\mathcal{T}_Y = \{U \mid U = V \cap Y \text{ for some } V \in \mathcal{T}\}
\]
is a topology on \(Y\) called the **subspace topology**. It is also called the **relative topology** on \(Y\) inherited from \(X\). The space \((Y, \mathcal{T}_Y)\) is called a (topological) **subspace** of \(X\). If \(U \in \mathcal{T}_Y\) we say \(U\) is **open in \(Y\)**.
**Theorem 3.25.** Let \((X, \mathcal{T})\) be a topological space and \(Y \subseteq X\). Then the collection of sets \(\mathcal{T}_Y\) is in fact a topology on \(Y\).
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