Discussion
Consider \(\mathbf{R}\) with the metric defined by :
\[ d(x, y) = \begin{cases} 1, & \text{if } x \ne y \\ 0, & \text{if } x = y \end{cases} \]
Which of the following statements is true for \((\mathbf{R}, d)\) ?
Answer: C
The true statement is [0, 1] is not compact.
In the discrete metric every singleton \(\{x\}\) is the open ball of radius 1/2 about \(x\). The singletons of [0, 1] form an open cover with no finite subcover, because [0, 1] is infinite. So [0, 1] is not compact; in a discrete space only finite sets are compact.
The others are false: every subset is both open and closed, so (0, 1) is closed and {1, ..., 100} is open; and every set is bounded, since no two points are more than 1 apart, so \((0, \infty)\) is bounded.
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