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d. S= {(x,0) CR²|-1 T= {(0, y) CR²|-1 ≤ x ≤ 1}. ≤ y ≤ 1}.

d. S= {(x,0) CR²|-1 T= {(0, y) CR²|-1 ≤ x ≤ 1}. ≤ y ≤ 1}.

Advanced Engineering Mathematics
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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6 d
**Problem 5** For each of the following pairs of shapes \( S \) and \( T \), sketch \( S \) and \( T \) and state whether they are homeomorphic. (Optional: If they are homeomorphic.) *Hint: Recall that the union of two sets \( A \) and \( B \) is the set consisting of exactly those elements which are contained in either \( A \) or \( B \).* a. \( S \subseteq \mathbb{R}^2 \) is the union of two circles centered at the origin of radii 1 and 2. \( T \) is the union of two circles centered at the origin of radii 3 and 4. b. \( S \) as in the previous problem. \( T \subseteq \mathbb{R}^2 \) is the union of a circle of radius 1 centered at the origin and a circle of radius 1 centered at \((0, 3)\). c. \( S \) as in the previous problem. \( T \subseteq \mathbb{R}^2 \) is the union of a circle of radius 1 centered at the origin and a circle of radius 1 centered at \((0, 1)\). d. \( S = \{(x, 0) \in \mathbb{R}^2 \,|\, -1 \leq x \leq 1\} \), \( T = \{(0, y) \in \mathbb{R}^2 \,|\, -1 \leq y \leq 1\} \).
Transcribed Image Text:**Problem 5** For each of the following pairs of shapes \( S \) and \( T \), sketch \( S \) and \( T \) and state whether they are homeomorphic. (Optional: If they are homeomorphic.) *Hint: Recall that the union of two sets \( A \) and \( B \) is the set consisting of exactly those elements which are contained in either \( A \) or \( B \).* a. \( S \subseteq \mathbb{R}^2 \) is the union of two circles centered at the origin of radii 1 and 2. \( T \) is the union of two circles centered at the origin of radii 3 and 4. b. \( S \) as in the previous problem. \( T \subseteq \mathbb{R}^2 \) is the union of a circle of radius 1 centered at the origin and a circle of radius 1 centered at \((0, 3)\). c. \( S \) as in the previous problem. \( T \subseteq \mathbb{R}^2 \) is the union of a circle of radius 1 centered at the origin and a circle of radius 1 centered at \((0, 1)\). d. \( S = \{(x, 0) \in \mathbb{R}^2 \,|\, -1 \leq x \leq 1\} \), \( T = \{(0, y) \in \mathbb{R}^2 \,|\, -1 \leq y \leq 1\} \).
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