Haha, that is the hard part of the problem if I'm interpreting your question correctly. Here's a discussion of the problem.
In my book they go through this proof of the following:
An open set is connected iff for any two points in a polygon from to lying entirely in .
We suppose satisfies this condition and is not connected to obtain a contradiction.
Pf:
Know where are both open and closed, and are nonempty.
If we let and
a polygon from to such that
We can assume
Define:
It can be shown that and are both open, contradicting the connectedness of .
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Now, I do not see how S and T are both open. Can anyone give me a hint please? Thank you.
-Sheld
Thanks for the link. I think that post thought is talking about the other direction of this iff . I was just wondering how do I show that the sets S and T, as described above, are open?
I think I have made progress!
Since is open,
So if we make epsilon sufficiently small,
however I'm not sure how small needs to be? Do I need to know? If we keep making the ball smaller and smaller, we end up with points only in , right?
You are right, my proof does make zero sense.
I can see how that function is continuous. The metric of G is absolute value? G is a subset of the complex numbers. And that theroem on continuity is very handy.
However since this is several chapters before continuity, I wonder if this can be proved without such heavy machinery.
Thank you very much for helping.