The discussion revolves around a conjecture regarding open sets in topology, specifically addressing the claim that if K is a union of subsets of G and K is open, then each subset in the union must also be open. Participants are exploring the validity of this conjecture and its implications in various topological contexts.
The discussion is active, with participants sharing counterexamples and questioning the original statement. There is an acknowledgment of the conjecture's falsehood, but no consensus on a definitive resolution has been reached. Various interpretations and examples are being explored.
Participants are discussing the implications of the conjecture in the context of different topologies, particularly focusing on non-discrete topologies where the conjecture does not hold. There is a recognition of the need for clarity regarding the definitions of open sets in these contexts.
quasar987 said:Or consider the classic example where one takes the reunion of the non-opens sets [1/n,+infty) and get the open sets (0,+infty)
morphism said:How do you expect to see the proof if you already know that the statement is false?!