Question

In the following, all subsets of Euclidean spaces are considered with the respective subspace topologies. Define an equivalence relation \( \sim \) on the sphere \[ S = \left\{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_1^2 + x_2^2 + x_3^2 = 1 \right\} \] by \( (x_1, x_2, x_3) \sim (y_1, y_2, y_3) \) if \( x_3 = y_3 \), for \( (x_1, x_2, x_3), (y_1, y_2, y_3) \in S \). Let \( [x_1, x_2, x_3] \) denote the equivalence class of \( (x_1, x_2, x_3) \), and let \( X \) denote the set of all such equivalence classes. Let \( L : S \to X \) be given by \[ L\left( (x_1, x_2, x_3) \right) = [x_1, x_2, x_3]. \] If \( X \) is provided with the quotient topology induced by the map \( L \), then which one of the following is TRUE?

• A. \( X \) is homeomorphic to \( \{ x \in \mathbb{R} : -1 \leq x \leq 1 \} \)
• B. \( X \) is homeomorphic to \( \{ (x_1, x_2) \in \mathbb{R}^2 : x_1^2 + x_2^2 = 1 \} \)
• C. \( X \) is homeomorphic to \( \{ (x_1, x_2) \in \mathbb{R}^2 : x_1^2 + x_2^2 \leq 1 \} \)
• D. \( X \) is homeomorphic to \( \{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_1^2 + x_2^2 = 1 { and } -1 \leq x_3 \leq 1 \} \)

Hint:

For quotient spaces induced by an equivalence relation, look for how the relation reduces the dimensions or "collapses" parts of the space. In this case, the quotient space corresponds to a line segment.

Answer 1

The Correct Option is A

We are given a quotient space \( X \) formed by an equivalence relation \( \sim \) on the sphere \( S \).
The equivalence relation \( (x_1, x_2, x_3) \sim (y_1, y_2, y_3) \) holds if and only if \( x_3 = y_3 \).

This means that for each \( x_3 \), there is a corresponding circle \( x_1^2 + x_2^2 = 1 - x_3^2 \) on the sphere.
As we vary \( x_3 \) from \( -1 \) to \( 1 \), we trace out a segment of a line.

Thus, the quotient space \( X \) corresponds to the interval \( [-1, 1] \), which is homeomorphic to \( \{ x \in \mathbb{R} : -1 \leq x \leq 1 \} \).

Final Answer
\[ \boxed{(A) \quad X \text{ is homeomorphic to } \{ x \in \mathbb{R} : -1 \leq x \leq 1 \}} \]