simply connecting curves

daku420

i know if every simple closed curve in D can be contracted to a point it is simply connected as in the case of|R^2 Domain or R^3 it is simply connected

but i am not feeling uncomfortable with |R^3 especially with the domains like when x =! 0 and y =! 0 why it is not simply connected?

how do I should visualize a closed curve with the domain in 3D?

Tinyboss

So R^3 is just the regular three-dimensional space we're all used to. If your domain is {(x,y,z) : x != 0 and y != 0}, then what does it look like? It's R^3, but with a line taken out of it--because the points that we took out look like (0,0,z), for whatever value of z. In fact we've removed the z-axis, because that's exactly where x=y=0.

Now imagine a circle that goes around that line. The line is infinite in both directions. How can you shrink that circle to a point, while never touching the line?

This is not a proof of course, but hopefully it can guide your intuition.