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I have a question regarding my solution to a problem in topology.

Problem:Show that if U is an open connected subset of ℝ^{2}, then U is also path-connected.

Hint: Show that given any x_{0}in U, show that the set of points that can be joined to x_{0}by a path in U is both open and closed.

First of all, I know that since U is connected, the only sets which are both open and closed are ∅ and U itself. Therefore, if I can show that the set of points which can be joined to x_{0}by a path is indeed open and closed, and non-empty, it is U. But I am stuck, and think I found an alternative solution to this problem, not using the hint, and was wondering if it is correct or not.

My idea is this. Since U is an open subset of ℝ^{2}, U can be written as a (possibly infinite) union of open balls B_{n}(x,ε), centered at x and with radius ε, positive, where x lies in U. And U is connected, which implies that given any ball, there is another ball with nonempty intersection. If it was empty, then U could be separated and hence not connected.

Now there is a path between any two pair of balls, which can be shown to imply that there is a path between any two points in the union of open balls.

It is very informal but do you guys think it could work? If not, any tips on how I find the set of all points in U connected to x_{0}by a path in U as the hint suggests?

Thanks in advance!

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# Show that an open connected subset of R^2 is path-connected

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