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?
Show Hint
- \( X \) is homeomorphic to \( \{ x \in \mathbb{R} : -1 \leq x \leq 1 \} \)
- \( X \) is homeomorphic to \( \{ (x_1, x_2) \in \mathbb{R}^2 : x_1^2 + x_2^2 = 1 \} \)
- \( X \) is homeomorphic to \( \{ (x_1, x_2) \in \mathbb{R}^2 : x_1^2 + x_2^2 \leq 1 \} \)
- \( X \) is homeomorphic to \( \{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_1^2 + x_2^2 = 1 { and } -1 \leq x_3 \leq 1 \} \)
The Correct Option is A
Solution and Explanation
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
\[ \boxed{(A) \quad X \text{ is homeomorphic to } \{ x \in \mathbb{R} : -1 \leq x \leq 1 \}} \]
Top Questions on Subsets
- Let \( E \subset F \) and \( F \subset K \) be field extensions which are not algebraic. Let \( \alpha \in K \) be algebraic over \( F \) and \( \alpha \notin F \). Let \( L \) be the subfield of \( K \) generated over \( E \) by the coefficients of the monic polynomial of minimal degree over \( F \) which has \( \alpha \) as a zero. Then, which of the following is/are TRUE?
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P1: \( R \) is isomorphic to the product of two rings \( R_1 \) and \( R_2 \).
P2: \( \exists r_1, r_2 \in R \) such that \( r_1^2 = r_1 \neq 0 \), \( r_2^2 = 0 \), and \( r_1 + r_2 = 1 \).
P3: \( R \) has ideals \( I_1, I_2 \subset R \) with \( R \neq I_1 \), \( (0) \neq I_2 \), and \( R = I_1 + I_2 \) and \( I_1 \cap I_2 = (0) \).
P4: \( \exists a, b \in R \) with \( a \neq 0 \), \( b \neq 0 \) such that \( ab = 0 \).
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Let the functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R}^2 \to \mathbb{R} \) be given by \[ f(x_1, x_2) = x_1^2 + x_2^2 - 2x_1x_2, \quad g(x_1, x_2) = 2x_1^2 + 2x_2^2 - x_1x_2. \] Consider the following statements:
S1: For every compact subset \( K \) of \( \mathbb{R} \), \( f^{-1}(K) \) is compact.
S2: For every compact subset \( K \) of \( \mathbb{R} \), \( g^{-1}(K) \) is compact. Then, which one of the following is correct?Consider the following subsets of the Euclidean space \( \mathbb{R}^4 \):
\( S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} \),
\( T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} \),
\( U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} \).
Then, which one of the following is TRUE?
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