The discussion revolves around the connectedness of subsets in R^n, specifically examining the claim that if a set S is connected, then its interior is also connected. Participants are exploring the validity of this statement and seeking to understand its implications.
The discussion is ongoing, with various perspectives being shared. Some participants suggest that the backward approach may provide insights, while others express uncertainty about how to construct a proof or counterexample. There is a recognition that simple examples may not suffice for generalization.
Participants note the challenge of generalizing from specific examples and the potential limitations of the original claim. There is an emphasis on the need for clarity regarding the definitions and properties of connectedness in the context of subsets of R^n.
kingwinner said:"Let S be a subset of R^n.
If S is connected, then the interior of S is connected."
Is this true or not?
I can't think of a counterexample, but I don't know how to prove it either...
?? All of [a,b], [a,b), (a,b], (a,b) are connected!loop quantum gravity said:I think you can take something like
an interval (a,b] or [a,b) (the same should work also for R^n).
obviously if [a,b) is connected, then (a,b) is'nt necessarily.