Discussion
Which of the following subsets of \(\mathbf{R}^{2}\) are not compact ?
(i) a circle
(ii) a parabola
(iii) \(S = \left\{(x, y) \in \mathbf{R}^{2}/y = \sin\dfrac{1}{x}, 0 \lt x \le 1\right\}\)
(iv) \(T = \{(x, y) \in \mathbf{R}^{2}/|x| + |y| = 4\}\)
Answer: B
The non-compact sets are (ii) and (iii).
In \(\mathbf{R}^{2}\), compact means closed and bounded (Heine-Borel). A parabola is closed but unbounded, so not compact. S, the graph of \(\sin(1/x)\) on \((0, 1]\), is bounded but not closed: its closure adds the segment from \((0, -1)\) to \((0, 1)\), which S does not contain.
A circle is closed and bounded, so compact. T, \(|x| + |y| = 4\), is the boundary of a square (a diamond) through \((\pm 4, 0)\) and \((0, \pm 4)\): closed and bounded, so also compact. That rules out every option containing (i) or (iv).
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