To give the examples we can use the following fact: the compact subsets of Rn are exactly the bounded closed sets.
Therefore the examples of compact curves would be:
- finite segments [a;b] for two points a,b∈R2,
- circles (x−a)2+(y−b)2=r2 ,
- finite segments of curves, i.e., C∩Bclosed(0,r) for any closed curve C and any r>0,
- an image of [0;1]⊆R by any continuous function f:[0;1]→R2.
And the examples of noncompact curves would respectively be:
- straight lines (such as y=kx+b),
- parabolas y=ax2+bx+c with a=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.)
Comments