Let's break down the homeomorphism \( C \times \mathbb{R}^+ = \mathbb{R}^+ \times S^1 \) and its components step by step.


### 1. **Understanding the Spaces:**


- **\( C \times \mathbb{R}^+ \):**

  - \( C \) represents the **complex plane**, which is the set of all complex numbers. This can be written as \( C = \mathbb{C} \), and it is a 2-dimensional space over the real numbers \( \mathbb{R} \).

  - \( \mathbb{R}^+ \) is the set of **positive real numbers**, \( \mathbb{R}^+ = (0, \infty) \). The product \( C \times \mathbb{R}^+ \) is then the Cartesian product of these two spaces, which can be thought of as a space that pairs each complex number with a positive real number.


- **\( \mathbb{R}^+ \times S^1 \):**

  - \( \mathbb{R}^+ \) again represents the positive real numbers.

  - \( S^1 \) is the **unit circle** in the complex plane. The unit circle consists of all complex numbers with magnitude 1, i.e., \( S^1 = \{ e^{i\theta} : \theta \in [0, 2\pi) \} \). The product space \( \mathbb{R}^+ \times S^1 \) can be thought of as a cylinder, where the radial distance is described by \( \mathbb{R}^+ \), and the angular coordinate (angle around the unit circle) is described by \( S^1 \).


### 2. **Homeomorphism:**

A **homeomorphism** is a continuous function between two topological spaces that has a continuous inverse. In simpler terms, two spaces are homeomorphic if you can "stretch" or "bend" one into the other without tearing or gluing. A homeomorphism preserves the topological structure of the spaces, meaning that the two spaces are "topologically equivalent."


The equation \( C \times \mathbb{R}^+ = \mathbb{R}^+ \times S^1 \) suggests that these two spaces are homeomorphic.


### 3. **Constructing the Homeomorphism:**

We can construct a homeomorphism \( \varphi : C \times \mathbb{R}^+ \to \mathbb{R}^+ \times S^1 \) using **polar coordinates**.


1. **Complex Number in Polar Form:**

   - Any complex number \( z \in C \) can be written in **polar form**: \( z = \rho e^{i\theta} \), where:

     - \( \rho = |z| \) is the magnitude of \( z \) (the distance from the origin),

     - \( \theta = \arg(z) \) is the argument (the angle of \( z \) in the complex plane).

   

2. **Define the Mapping \( \varphi \):**

   We define the map \( \varphi : C \times \mathbb{R}^+ \to \mathbb{R}^+ \times S^1 \) as follows:

   \[

   \varphi(z, r) = (r \cdot \rho, e^{i\theta})

   \]

   where:

   - \( z = \rho e^{i\theta} \in C \) is the complex number in polar form,

   - \( r \in \mathbb{R}^+ \) is the positive real number.


3. **Interpretation of the Mapping:**

   - \( r \cdot \rho \in \mathbb{R}^+ \) represents the radial distance in the cylindrical coordinate system (combining the distance \( \rho \) from the complex plane and \( r \) from \( \mathbb{R}^+ \)).

   - \( e^{i\theta} \in S^1 \) represents the angular coordinate on the unit circle, preserving the angle \( \theta \).


### 4. **Verification of the Homeomorphism:**

To verify that \( \varphi \) is indeed a homeomorphism, we need to check two main properties:


- **Bijectivity:** 

  - **Injectivity (One-to-one):** For each pair \( (z, r) \in C \times \mathbb{R}^+ \), the map \( \varphi \) gives a unique point in \( \mathbb{R}^+ \times S^1 \). Since \( \rho \) and \( \theta \) uniquely determine \( z \), and \( r \) is explicitly part of the map, \( \varphi \) is injective.

  - **Surjectivity (Onto):** Every point in \( \mathbb{R}^+ \times S^1 \) can be reached by some point in \( C \times \mathbb{R}^+ \). Given any \( (r', e^{i\theta'}) \in \mathbb{R}^+ \times S^1 \), we can always find \( z = \rho e^{i\theta'} \) and map it accordingly, making \( \varphi \) surjective.

  

- **Continuity:** 

  - The map \( \varphi \) is continuous because both the polar coordinate transformation and multiplication by \( r \) are continuous operations.

  - The inverse map is also continuous. To invert \( \varphi \), we would simply reverse the process: extract the modulus \( r \cdot \rho \) and angle \( \theta \), then recover \( z \) in the complex plane.


### 5. **Conclusion:**


The map \( \varphi : C \times \mathbb{R}^+ \to \mathbb{R}^+ \times S^1 \), defined by

\[

\varphi(z, r) = (r \cdot \rho, e^{i\theta}),

\]

is a **homeomorphism** because it is a continuous bijection with a continuous inverse. This demonstrates that the space \( C \times \mathbb{R}^+ \) (the complex plane with positive real numbers) is topologically equivalent to \( \mathbb{R}^+ \times S^1 \) (a cylinder), even though their representations may initially seem different. The homeomorphism allows one space to be continuously transformed into the other, preserving their topological properties.