The discussion revolves around a problem in real analysis and topology, specifically examining the properties of a non-empty subset A of ℝ, where both A and its complement are open subsets of ℝ. Participants are tasked with proving various properties of A, including its boundedness and the implications of its openness.
The discussion is ongoing, with participants providing insights and questioning assumptions. Some have suggested reconsidering the implications of certain assumptions, while others are exploring the relationship between the supremum of A and its membership in A or its complement.
There is an emphasis on the properties of open sets in ℝ and the implications of connectedness in the context of the problem. Participants are also navigating the constraints of the problem statement and the requirements for their proofs.
vela said:Why are you saying ##m## is inf(B)? The problem statement only says to show that ##m## bounds B from below and is greater than or equal to ##x##. It's not necessarily the greatest lower bound.