An open connected set is path(polygon) connected

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SUMMARY

The discussion centers on the proof of path connectivity in open connected sets, specifically referencing the text "Advanced Calculus" by R. Creighton Buck. The user proposes that by utilizing open balls within a non-empty open connected set O, one can demonstrate that every point in O can be connected through a sequence of straight lines. The proof hinges on the definition of connectedness and the properties of open sets, emphasizing the necessity of ensuring that the entirety of O can be covered by these open balls.

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  • Understanding of open connected sets in topology
  • Familiarity with the concept of path connectivity
  • Knowledge of open balls in metric spaces
  • Basic principles of calculus and mathematical proofs
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  • Study the properties of open connected sets in topology
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  • Explore the use of open balls in proofs related to connectedness
  • Review advanced calculus concepts as presented in "Advanced Calculus" by R. Creighton Buck
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Mathematics students, particularly those studying topology and advanced calculus, as well as educators seeking to clarify concepts of connectedness and path connectivity in open sets.

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.
 

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