M Problem 5. For each of the following pairs of shapes S a and state whether they are homeomorphic. (Optional: phic, also say whether they are isotopic.) HINT: Recall sets A and B is the set consisting of exactly those elemer in either A or B. 9 02nd blow to a. SCR2 is the union of two circles centered centere 1 and 2. TC R2 is the union of two circles centere of radii 3 and 4. b. S as in the previous problem. TCR2 is the union of a circle of radius 1 cent circle of radius 1 at centered at (0,3). 098A SA c. S as in the previous problem. TCR2 is the union of a circle of radius 1 cen circle of radius 1 at centered at (0, 1). M d. S = {(x,0) C R² | -1 ≤ x ≤ 1}. T = {(0, y) CR²| -1 ≤ y ≤ 1}.
M Problem 5. For each of the following pairs of shapes S a and state whether they are homeomorphic. (Optional: phic, also say whether they are isotopic.) HINT: Recall sets A and B is the set consisting of exactly those elemer in either A or B. 9 02nd blow to a. SCR2 is the union of two circles centered centere 1 and 2. TC R2 is the union of two circles centere of radii 3 and 4. b. S as in the previous problem. TCR2 is the union of a circle of radius 1 cent circle of radius 1 at centered at (0,3). 098A SA c. S as in the previous problem. TCR2 is the union of a circle of radius 1 cen circle of radius 1 at centered at (0, 1). M d. S = {(x,0) C R² | -1 ≤ x ≤ 1}. T = {(0, y) CR²| -1 ≤ y ≤ 1}.
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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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, also say whether they are isotopic.)
**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. \( S \subset \mathbb{R}^2 \) is the union of two circles centered at the origin, of radii 1 and 2. \( T \subset \mathbb{R}^2 \) is the union of two circles centered at the origin, of radii 3 and 4.
b. \( S \) as in the previous problem. \( T \subset \mathbb{R}^2 \) is the union of a circle of radius 1 centered at the origin and a circle of radius 1 at \((0, 3)\).
c. \( S \) as in the previous problem. \( T \subset \mathbb{R}^2 \) is the union of two circles centered at \((1, 0)\) and \((0, 1)\).
d. \( S = \{(x, 0) \mid -1 \leq x \leq 1 \}\), \( T = \{(0, y) \mid -1 \leq y \leq 1 \} \).
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