- #1

lugita15

- 1,554

- 15

^{2}look like a bunch of regions bounded by continuous curves, or are there open sets with weirder boundaries than that? Let me state my question more formally.

A Jordan curve is a continuous closed curve in ℝ

^{2}without self-intersections. The Jordan curve theorem states that the complement of any Jordan curve has two connected components, an interior and an exterior.

Now let's define a simple unbounded curve to be a continuous map f: (-∞,∞) → ℝ

^{2}such that f((-∞,0)) and f((0,∞)) are both unbounded. Then does the complement of a simple unbounded curve always have two connected components? It seems intuitively true, since you'd expect curve to have two sides, but considering how long it took to prove the Jordan curve theorem, things may not be as straightforward as they appear.

Assuming it is true, let us call a "side" any connected component of a complement of a continuous curve in ℝ

^{2}. (Note that some continuous curves in ℝ

^{2}have a connected complement.) Then my question is, can any open set U in ℝ

^{2}be written as a countable union of disjoint open sets U_1, U_2, ..., such that each U_i is either a side or an intersection of two sides? If not, what if we let each U_i be a finite intersection of sides, rather than just an intersection of two sides?

Can this be generalized to higher dimensions?

Any help would be greatly appreciated.

Thank You in Advance.