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1. How would I go about proving that an annulus in R^2 is path connected? It seems like a mess. I tried to make 4 central points (a+b/2,0), (0, a+b/2), (-(a+b/2),0), (0, -(a+b/2)) (where the annulus is {(x,y) : a^2 < x^2 + y^2 < b^2} and show that these 4 points are path connected and then "obviously" any point in the annulus can be path connected to one of these central points. But this seemed too fiddly and difficult.
2. (Original post by gangsta316)
How would I go about proving that an annulus in R^2 is path connected? It seems like a mess. I tried to make 4 central points (a+b/2,0), (0, a+b/2), (-(a+b/2),0), (0, -(a+b/2)) (where the annulus is {(x,y) : a^2 < x^2 + y^2 < b^2} and show that these 4 points are path connected and then "obviously" any point in the annulus can be path connected to one of these central points. But this seemed too fiddly and difficult.
It's not obvious that any point can be path connected to one of those four. (well... it is... but the proof seems about as hard as the general case).

I'd do it directly. If you think of R^2 as the complex numbers (just so I can use the concept of an argument), have the argument and modulus vary as some parameter t goes from 0 to 1. So if P1 and P2 are points in the annulus, when t = 0, you're at point P1, and when t=1 you're at point P2.

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