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?
To determine the connectedness of the given subsets of \( \mathbb{R}^4 \), we need to explore the nature of each set individually:
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.
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.
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:
Thus, the correct statement is: \( S \) and \( T \) are connected, but \( U \) is not connected.
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?