The conjecture stating that if K is a union of subsets of G with K open, then each subset in the union is open, has been proven false. Participants in the discussion provided counterexamples, such as the union of the non-open sets [1/n, +∞) resulting in the open set (0, +∞). Additionally, the union of the point 0 and the interval (-1, 1) produces the open set (-1, 1), while the point 0 remains closed. The consensus is that this conjecture does not hold in non-discrete topologies.
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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?!