Talk:Knaster–Kuratowski fan
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Proof
[edit]It would be great to include a proof that the fan is connected, and that it is totally disconnected when its apex is removed. For the benefit of the visual thinkers among us, it would be good to get a picture—not of the set itself, of course, but some notion of its construction.Dfeuer (talk) 16:52, 1 October 2011 (UTC)
- Total disconnectedness is simple enough to prove at least. The Cantor set is itself totally disconnected, and indeed ∀ c₁, c₂ ∈ C, c₁ ≠ c₂: ∃ [d, e] ∉ C: c₁ < d < e < c₂ or c₁ > d > e > c₂. So for any two rays in the leaky tent, there are lines through (d, 0) and p, likewise (e, 0) and p separating them. By construction each ray is also individually totally disconnected. --Trɔpʏliʊm • blah 22:39, 29 December 2019 (UTC)