Question

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?

• A. \( S \) is connected, but \( T \) and \( U \) are not connected.
• B. \( T \) and \( U \) are connected, but \( S \) is not connected.
• C. \( S \) and \( U \) are connected, but \( T \) is not connected.
• D. \( S \) and \( T \) are connected, but \( U \) is not connected.

Hint:

For quadratic forms in Euclidean spaces, if the equation represents a hyperboloid of one sheet, the set is connected. If it represents a hyperboloid of two sheets, the set is disconnected.

Answer 1

The Correct Option is D

To determine the connectedness of the given subsets of \( \mathbb{R}^4 \), we need to explore the nature of each set individually:

1. Set \( S \):

   Defined as \( S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} \).

   This represents a cone in four-dimensional space, as it can be rearranged to x_4^2 = x_1^2 + x_2^2 + x_3^2. Such a set is continuous and unbounded around the origin, without any breaks, indicating it is connected.

2. Set \( T \):

   Defined as \( T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} \).

   This represents a hyperboloid of one sheet, which is known to be connected due to its smooth and continuous nature without any disjoint components.

3. Set \( U \):

   Defined as \( U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} \).

   This represents a hyperboloid of two sheets, which exists in two disjoint parts. Because of this separation, the set \( U \) is not connected.

By evaluating each set, we conclude that:

• Set \( S \) is connected.
• Set \( T \) is connected.
• Set \( U \) is not connected.

Thus, the correct statement is: \( S \) and \( T \) are connected, but \( U \) is not connected.