Question

Consider the function \( f : \mathbb{R}^2 \to \mathbb{R}^2 \) given by \[ f(x, y) = (e^{2\pi x} \cos 2\pi y, e^{2\pi x} \sin 2\pi y). \] Then, which of the following is/are TRUE?

• A. If \( G \) is open in \( \mathbb{R}^2 \), then \( f(G) \) is open in \( \mathbb{R}^2 \)
• B. If \( G \) is closed in \( \mathbb{R}^2 \), then \( f(G) \) is closed in \( \mathbb{R}^2 \)
• C. If \( G \) is dense in \( \mathbb{R}^2 \), then \( f(G) \) is dense in \( \mathbb{R}^2 \)
• D. \( f \) is surjective

Hint:

In general, differentiable maps preserve the openness and density of sets, but they do not always preserve closedness or surjectivity.

Answer 1

The Correct Option is A, C

The function \( f(x, y) = (e^{2\pi x} \cos 2\pi y, e^{2\pi x} \sin 2\pi y) \) maps points from \( \mathbb{R}^2 \) to another \( \mathbb{R}^2 \), and it is continuous and differentiable.

Step 1: Openness of \( f(G) \) when \( G \) is open
Since \( f \) is a smooth and differentiable function, the image of an open set under \( f \) will also be open. This is a general result for differentiable maps in topology. Therefore, (A) is TRUE.

Step 2: Density of \( f(G) \) when \( G \) is dense
Since \( f \) is a continuous map and the set \( G \) is dense in \( \mathbb{R}^2 \), it follows that \( f(G) \) will be dense in \( \mathbb{R}^2 \) as well, as continuous maps preserve density. Therefore, (C) is TRUE.

Step 3: Closedness of \( f(G) \) when \( G \) is closed
The map \( f \) is not guaranteed to preserve closedness because the exponential function can map closed sets to non-closed sets. Hence, (B) is FALSE.

Step 4: Surjectivity of \( f \)
The function \( f \) is not surjective because its range is a subset of the plane \( \mathbb{R}^2 \), specifically a subset that excludes points where the radial distance (from the origin) is 0. Thus, (D) is FALSE.

Final Answer
\[ \boxed{A} \quad \text{If } G \text{ is open in } \mathbb{R}^2, \text{ then } f(G) \text{ is open in } \mathbb{R}^2 \]
\[ \boxed{C} \quad \text{If } G \text{ is dense in } \mathbb{R}^2, \text{ then } f(G) \text{ is dense in } \mathbb{R}^2 \]