An open connected set is path(polygon) connected

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In summary, the conversation discusses a proof in the textbook "Advanced Calculus" and the idea of connecting points in an open set through intersecting subsets. The speaker is seeking clarification and direction in their understanding of filling up an open set with open balls and connecting points through straight lines. They have also consulted with a professor for further clarification.
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kidmode01
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Hello there, can someone help me with the proof? The proof in my text (Advanced Calculus, R Creighton Buck) is long and tedious and I hoped to be able to make it shorter

Let O be a non empty open connected set.

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Aside:
Then if O is broken up into Kn sets, then the intersection of some Kn with some other Kn is nonempty just by the definition of connectedness.
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So I guess my idea is that we can make all these Kn's into open balls(or some sets) and connect every point through the intersection of the Kn's.

Consider an open ball completely contained in O. Since any point 'p' of an open set must not be isolated, there exists another point 'q' such that p and q can be connected by a straight line inside the ball.

Then by my "idea" every point in O can be connected by a sequence of straight lines.

Could someone point me in the right direction if I'm off? The thing I'm having trouble wrapping my head around is filling up all of O with open balls. It seems this only works in some finite sense or something is mucked up.
 
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I went and talked to one of my prof's about it, got it staightened out.
 

What is an open connected set?

An open connected set is a set of points in a topological space that is both open (meaning every point has a neighborhood contained within the set) and connected (meaning there is a continuous path connecting any two points in the set).

What does it mean for an open connected set to be path connected?

A path connected open set is one where there exists a continuous function from a closed interval [a,b] in the real numbers to the set such that the starting point of the path is a and the ending point is b. In other words, there is a path from any point to any other point in the set.

Why is path connectedness important in open connected sets?

Path connectedness is important in open connected sets because it ensures that there are no "holes" or gaps in the set. This means that the set is "connected" in the sense that every point is reachable from any other point via a continuous path.

Can an open connected set be path disconnected?

No, an open connected set cannot be path disconnected. If it were, then there would exist two points in the set that are not connected by a path, which goes against the definition of open connectedness.

What are some examples of open connected sets that are path connected?

Some examples of open connected sets that are path connected include a circle, a sphere, and a disk in two or three dimensions. In general, any set that is "smooth" and has no holes or gaps can be considered path connected.

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