Let S² be the sphere, D the disk, T the torus, S' the circle, and I = [0, 1] with the standard topology. Draw pictures of the product spaces S2 x I, T × I, S' × I x I, and S' × D. If M is a mobius band what does M × I look like?
Let S² be the sphere, D the disk, T the torus, S' the circle, and I = [0, 1] with the standard topology. Draw pictures of the product spaces S2 x I, T × I, S' × I x I, and S' × D. If M is a mobius band what does M × I look like?
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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![**Transcription:**
Let \( S^2 \) be the sphere, \( D \) the disk, \( T \) the torus, \( S^1 \) the circle, and \( I = [0,1] \) with the standard topology. Draw pictures of the product spaces \( S^2 \times I \), \( T \times I \), \( S^1 \times I \times I \), and \( S^1 \times D \). If \( M \) is a mobius band what does \( M \times I \) look like?
**Explanation of Concepts:**
1. **Product Spaces:**
- **\( S^2 \times I \):** This represents a cylinder-like shape where each slice is a sphere.
- **\( T \times I \):** This describes a thickened torus, or a torus 'tube.'
- **\( S^1 \times I \times I \):** This forms a solid torus, visualized as a donut shape.
- **\( S^1 \times D \):** This can be thought of as a solid torus as well.
2. **Möbius Band Product:**
- **\( M \times I \):** This would look like a three-dimensional strip with a 180-degree twist, extended into a ‘prism’ by multiplying with the interval \( I \).
These descriptions aim to visualize familiar topological forms and their product spaces by extending basic shapes in new dimensions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F36268169-052e-482b-acb5-c5dcae700f3f%2F6de6ff57-e1f3-4997-98a2-a35c09f97c35%2Fp8fcv1p_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription:**
Let \( S^2 \) be the sphere, \( D \) the disk, \( T \) the torus, \( S^1 \) the circle, and \( I = [0,1] \) with the standard topology. Draw pictures of the product spaces \( S^2 \times I \), \( T \times I \), \( S^1 \times I \times I \), and \( S^1 \times D \). If \( M \) is a mobius band what does \( M \times I \) look like?
**Explanation of Concepts:**
1. **Product Spaces:**
- **\( S^2 \times I \):** This represents a cylinder-like shape where each slice is a sphere.
- **\( T \times I \):** This describes a thickened torus, or a torus 'tube.'
- **\( S^1 \times I \times I \):** This forms a solid torus, visualized as a donut shape.
- **\( S^1 \times D \):** This can be thought of as a solid torus as well.
2. **Möbius Band Product:**
- **\( M \times I \):** This would look like a three-dimensional strip with a 180-degree twist, extended into a ‘prism’ by multiplying with the interval \( I \).
These descriptions aim to visualize familiar topological forms and their product spaces by extending basic shapes in new dimensions.
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