**Theorem 12.3** *Given topological spaces \(X\) and \(Y\) with \(A \subset X\), homotopy relative to \(A\) is an equivalence relation on the set of all continuous functions from \(X\) to \(Y\). In particular, if \(A = \emptyset\) (the empty set), homotopy is an equivalence relation on the set of all continuous functions from \(X\) to \(Y\).* **Theorem** (Fundamental Theorem of Algebra) *A polynomial \(p(z) = a_nz^n + \cdots + a_1z + a_0\) with complex coefficients and degree \(n \geq 1\) has at least one root.* This theorem does not seem at first to be a topological theorem! However, we can get our first hint that topology plays a role by looking at a special case. **Exercise 12.1** *A polynomial \(p(x) = a_nx^n + \cdots + a_1x + a_0\) with real coefficients where \(a_n \neq 0\) and \(n\) is odd has at least one real root.* **Definition** *Let \(X\) and \(Y\) be topological spaces, and let \(f, g : X \to Y\) be continuous functions. Then \(f\) is homotopic to \(g\) (written \(f \simeq g\)) if and only if there is a continuous map \(F : X \times [0, 1] \to Y\) such that the equations* \[ F(x, 0) = f(x) \] \[ F(x, 1) = g(x) \] *hold for all \(x \in X\). The map \(F\) is called a homotopy between \(f\) and \(g\).* **Definition** *A function \(f : X \to Y\) whose image is a single point is called a constant map. A map is said to be null homotopic if and only if it is homotopic to the constant map.* A null homotopic map is one in which the image \(f(X)\) in \(Y\) can be gradually deformed within \(Y\) to a map that takes
**Theorem 12.3** *Given topological spaces \(X\) and \(Y\) with \(A \subset X\), homotopy relative to \(A\) is an equivalence relation on the set of all continuous functions from \(X\) to \(Y\). In particular, if \(A = \emptyset\) (the empty set), homotopy is an equivalence relation on the set of all continuous functions from \(X\) to \(Y\).* **Theorem** (Fundamental Theorem of Algebra) *A polynomial \(p(z) = a_nz^n + \cdots + a_1z + a_0\) with complex coefficients and degree \(n \geq 1\) has at least one root.* This theorem does not seem at first to be a topological theorem! However, we can get our first hint that topology plays a role by looking at a special case. **Exercise 12.1** *A polynomial \(p(x) = a_nx^n + \cdots + a_1x + a_0\) with real coefficients where \(a_n \neq 0\) and \(n\) is odd has at least one real root.* **Definition** *Let \(X\) and \(Y\) be topological spaces, and let \(f, g : X \to Y\) be continuous functions. Then \(f\) is homotopic to \(g\) (written \(f \simeq g\)) if and only if there is a continuous map \(F : X \times [0, 1] \to Y\) such that the equations* \[ F(x, 0) = f(x) \] \[ F(x, 1) = g(x) \] *hold for all \(x \in X\). The map \(F\) is called a homotopy between \(f\) and \(g\).* **Definition** *A function \(f : X \to Y\) whose image is a single point is called a constant map. A map is said to be null homotopic if and only if it is homotopic to the constant map.* A null homotopic map is one in which the image \(f(X)\) in \(Y\) can be gradually deformed within \(Y\) to a map that takes
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ISBN:9780470458365
Author:Erwin Kreyszig
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Could you explain how to do 12.3 with detailed explanation? Thank you!
![**Theorem 12.3**
*Given topological spaces \(X\) and \(Y\) with \(A \subset X\), homotopy relative to \(A\) is an equivalence relation on the set of all continuous functions from \(X\) to \(Y\). In particular, if \(A = \emptyset\) (the empty set), homotopy is an equivalence relation on the set of all continuous functions from \(X\) to \(Y\).*
**Theorem** (Fundamental Theorem of Algebra)
*A polynomial \(p(z) = a_nz^n + \cdots + a_1z + a_0\) with complex coefficients and degree \(n \geq 1\) has at least one root.*
This theorem does not seem at first to be a topological theorem! However, we can get our first hint that topology plays a role by looking at a special case.
**Exercise 12.1**
*A polynomial \(p(x) = a_nx^n + \cdots + a_1x + a_0\) with real coefficients where \(a_n \neq 0\) and \(n\) is odd has at least one real root.*
**Definition**
*Let \(X\) and \(Y\) be topological spaces, and let \(f, g : X \to Y\) be continuous functions. Then \(f\) is homotopic to \(g\) (written \(f \simeq g\)) if and only if there is a continuous map \(F : X \times [0, 1] \to Y\) such that the equations*
\[
F(x, 0) = f(x)
\]
\[
F(x, 1) = g(x)
\]
*hold for all \(x \in X\). The map \(F\) is called a homotopy between \(f\) and \(g\).*
**Definition**
*A function \(f : X \to Y\) whose image is a single point is called a constant map. A map is said to be null homotopic if and only if it is homotopic to the constant map.*
A null homotopic map is one in which the image \(f(X)\) in \(Y\) can be gradually deformed within \(Y\) to a map that takes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56be79ad-be6a-45f1-b897-58d23fd7e62d%2F454075c1-61b3-4fb0-abf3-651c11bafae3%2Fmt57rib_processed.png&w=3840&q=75)
Transcribed Image Text:**Theorem 12.3**
*Given topological spaces \(X\) and \(Y\) with \(A \subset X\), homotopy relative to \(A\) is an equivalence relation on the set of all continuous functions from \(X\) to \(Y\). In particular, if \(A = \emptyset\) (the empty set), homotopy is an equivalence relation on the set of all continuous functions from \(X\) to \(Y\).*
**Theorem** (Fundamental Theorem of Algebra)
*A polynomial \(p(z) = a_nz^n + \cdots + a_1z + a_0\) with complex coefficients and degree \(n \geq 1\) has at least one root.*
This theorem does not seem at first to be a topological theorem! However, we can get our first hint that topology plays a role by looking at a special case.
**Exercise 12.1**
*A polynomial \(p(x) = a_nx^n + \cdots + a_1x + a_0\) with real coefficients where \(a_n \neq 0\) and \(n\) is odd has at least one real root.*
**Definition**
*Let \(X\) and \(Y\) be topological spaces, and let \(f, g : X \to Y\) be continuous functions. Then \(f\) is homotopic to \(g\) (written \(f \simeq g\)) if and only if there is a continuous map \(F : X \times [0, 1] \to Y\) such that the equations*
\[
F(x, 0) = f(x)
\]
\[
F(x, 1) = g(x)
\]
*hold for all \(x \in X\). The map \(F\) is called a homotopy between \(f\) and \(g\).*
**Definition**
*A function \(f : X \to Y\) whose image is a single point is called a constant map. A map is said to be null homotopic if and only if it is homotopic to the constant map.*
A null homotopic map is one in which the image \(f(X)\) in \(Y\) can be gradually deformed within \(Y\) to a map that takes
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