Answer to Question #213035 in Functional Analysis for smi

Question

Give the examples of compact and noncompact curves in the plane R^2.

Answer 1

To give the examples we can use the following fact: the compact subsets of $\mathbb{R}^n$ are exactly the bounded closed sets.

Therefore the examples of compact curves would be:

• finite segments $[a;b]$ for two points $a,b\in\mathbb{R}^2$,
• circles $(x-a)^2+(y-b)^2=r^2$ ,
• finite segments of curves, i.e., $\mathcal{C}\cap B_{closed}(0,r)$ for any closed curve $\mathcal{C}$ and any $r>0$,
• an image of $[0;1]\subseteq \mathbb{R}$ by any continuous function $f:[0;1]\to \mathbb{R}^2$.

And the examples of noncompact curves would respectively be:

• straight lines (such as $y=kx+b$),
• parabolas $y=ax^2+bx+c$ with $a\neq 0$ , hyperbolas $(y-a)^2-(x-b)^2=c$,
• any other unbounded curves (sinusoids, curves $y=P(x)$ with a non-constant polynomial $P$ etc.)
• not closed curves (circle minus a point, an open-ended ray, semi-open interval, open-ended branch of a parabola, etc.)