Question #213035

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


1
Expert's answer
2021-07-05T16:56:56-0400

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

Therefore the examples of compact curves would be:

  • finite segments [a;b][a;b] for two points a,bR2a,b\in\mathbb{R}^2,
  • circles (xa)2+(yb)2=r2(x-a)^2+(y-b)^2=r^2 ,
  • finite segments of curves, i.e., CBclosed(0,r)\mathcal{C}\cap B_{closed}(0,r) for any closed curve C\mathcal{C} and any r>0r>0,
  • an image of [0;1]R[0;1]\subseteq \mathbb{R} by any continuous function f:[0;1]R2f:[0;1]\to \mathbb{R}^2.

And the examples of noncompact curves would respectively be:

  • straight lines (such as y=kx+by=kx+b),
  • parabolas y=ax2+bx+cy=ax^2+bx+c with a0a\neq 0 , hyperbolas (ya)2(xb)2=c(y-a)^2-(x-b)^2=c,
  • any other unbounded curves (sinusoids, curves y=P(x)y=P(x) with a non-constant polynomial PP etc.)
  • not closed curves (circle minus a point, an open-ended ray, semi-open interval, open-ended branch of a parabola, etc.)

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