Question

Let \( \mathbb{R}^1 \) and \( \mathbb{R}^2 \) be provided with the respective Euclidean topologies, and let \[ S^1 = \{ (x_1, x_2) \in \mathbb{R}^2 : x_1^2 + x_2^2 = 1 \} \] be assigned the subspace topology induced from \( \mathbb{R}^2 \). If \( f: S^1 \to \mathbb{R}^1 \) is a non-constant continuous function, then which of the following is/are TRUE?

• A. \( f \) maps closed sets to closed sets
• B. \( f \) is injective
• C. \( f \) is surjective
• D. There exists \( \lambda \in \mathbb{R} \) such that \( f(\cos \lambda, \sin \lambda) = f(-\cos \lambda, -\sin \lambda) \)

Hint:

For continuous functions on compact sets, closed sets in the domain map to closed sets in the codomain. This property is key in many topological results.

Answer 1

The Correct Option is A, D

The function \( f: S^1 \to \mathbb{R}^1 \) is continuous and non-constant. We are asked to analyze its properties.

Step 1: \( f \) maps closed sets to closed sets
A continuous function on a compact space (such as \( S^1 \), the unit circle, which is compact in \( \mathbb{R}^2 \)) maps closed sets to closed sets. Therefore, (A) is TRUE.

Step 2: Injectivity of \( f \)
Since the function is non-constant, it cannot be injective on \( S^1 \), because multiple points on \( S^1 \) could map to the same value in \( \mathbb{R}^1 \). Thus, (B) is FALSE.

Step 3: Surjectivity of \( f \)
For a continuous function on the unit circle, \( f \) is not necessarily surjective onto \( \mathbb{R}^1 \), as \( S^1 \) is a compact set and the image of \( f \) could be a proper subset of \( \mathbb{R}^1 \). Therefore, (C) is FALSE.

Step 4: The existence of \( \lambda \)
Given that \( f \) is continuous and non-constant, by the properties of continuous functions on the circle, there exists a point \( \lambda \) where \( f(\cos \lambda, \sin \lambda) = f(-\cos \lambda, -\sin \lambda) \), because these points are diametrically opposite on the unit circle. Hence, (D) is TRUE.

Final Answer
\[ \boxed{(A) \quad f \text{ maps closed sets to closed sets}} \] \[ \boxed{(D) \quad \text{There exists } \lambda \in \mathbb{R} \text{ such that } f(\cos \lambda, \sin \lambda) = f(-\cos \lambda, -\sin \lambda)} \]