Question

In Chapter 3 we introduced three different representations of the torus: (i) The definition of the torus as a subspace of $\mathbb{R}^3$ in Example 3.5, (ii) The product space $S^1 \times S^1$ in Example 3.10, and (iii) A quotient space obtained by gluing edges of a square in Example 3.20. Verify that these spaces are topologically equivalent to each other by defining homeomorphisms between them.

   In Chapter 3 we introduced three different representations of the torus:
(i) The definition of the torus as a subspace of $\mathbb{R}^3$ in Example 3.5,
(ii) The product space $S^1 \times S^1$ in Example 3.10, and (iii) A quotient space obtained by gluing edges of a square in Example 3.20.

Verify that these spaces are topologically equivalent to each other by defining homeomorphisms between them.

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Introduction to Topology: Pure and Applied

Introduction to Topology: Pure and Applied

Colin Adams, Robert… 1st Edition

Chapter 4, Problem 35

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5, the torus \(T^2\) can be defined as the set of points in \(\mathbb{R}^3\) given by the equation: \[ T^2 = \{ (x, y, z) \in \mathbb{R}^3 \mid (x^2 + y^2 + z^2 + 1)^2 = 4z^2 \} \] This represents a toroidal shape in three-dimensional space. Show more…

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In Chapter 3 we introduced three different representations of the torus: (i) The definition of the torus as a subspace of $\mathbb{R}^3$ in Example 3.5, (ii) The product space $S^1 \times S^1$ in Example 3.10, and (iii) A quotient space obtained by gluing edges of a square in Example 3.20. Verify that these spaces are topologically equivalent to each other by defining homeomorphisms between them.

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