try and prove this...

[Keiji]

[Nick]

No; for every junction you engulf, you create two new junctions. Even if you go on forever, you can never engulf all of them.

EDIT: woops... misread the command. What kind of proof do you want? A derivation, or a flow proof?

[Keiji]

I see this topic died because I never replied to your question.

Hmm, I'm not sure what you mean by a flow proof. Any mathematical proof works for me, so long as it's understandable

[PWrong]

Those first two lines, are they infinitely long or just line segments? Do the ends of a line segment count as a junction if they don't touch anything?

I was hoping I might find a counterexample in hyperbolic space, or maybe using an infinite set of lines, but I can't find one.

I have a sketch of a proof for the euclidean case with a finite number of lines. Draw all the lines in blue pen. Now draw a red polygon that contains all the junctions so far, but you can only draw over the blue lines. Now to engulf a junction, we have to draw a line which must be outside the polygon. This line has to hit another line at some point. If it hits a line outside the polygon, then we have a new junction that must be engulfed. If the line ever reaches the polygon, then it must hit a blue line on the edge of the polygon. This still creates a new junction which touches the outside of the diagram, so that has to be engulfed. So either way, you can't engulf a junction without creating a new unengulfed junction.

Now at the moment this only works for euclidean space, because otherwise this line might not be true:

This line has to hit another line at some point.

Also, we can only draw a finite number of lines, otherwise we might not be able to draw a polygon around them.

[Keiji]

Those first two lines, are they infinitely long or just line segments? Do the ends of a line segment count as a junction if they don't touch anything?

The lines have finite length, and only intersections count as junctions.

In curved space the problem is trivial: considering my first diagram, connect the topmost line end to the bottommost line end using a straight line which wraps around the curved surface. If the surface is curved over on the right-hand side, it will engulf the three junctions. However, in flat space, it is impossible to do this. I'm not sure what happens in hyperbolic space.

[pat]

I think the concept "engulfed" has problems in compact (i.e. non-infinite) manifolds. No?

Assertion, there are lions on the planet. Problem: put at least one lion in a cage. Solution: step 1. Construct cage around yourself. step 2. Declare yourself on the outside.

[pat]

Put another way... consider the phrasing of the Jordan curve theorem:

Any simple, closed curve in the plane divides the plane into two regions a bounded region called the interior and an unbounded region called the exterior. The curve forms the boundary of each.

On a compact, two-dimensional manifold, there is no unbounded part even before you start dividing it up. So, if I draw a circle... which side is the inside and which side is the outside?