Turing's topological proof that every written alphabet is finite (2010)

Re:Turing's topological proof that every written alphabet is... by ridiculous_fish on Tuesday July 23, 2024 @05:56PM (#41051383)

Why do we need the property that X is compact? Say we define X (the square) to exclude the boundary: X = (0,1)x(0,1). This is no longer a compact set, but it seems silly for the proof to now fail.

Is boundedness sufficient?

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Re:Turing's topological proof that every written alphabet is... by NotYourLawyer on Tuesday July 23, 2024 @11:10PM (#41053302)

“Farthest” point doesn’t work if you drop compactness.

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Re:Turing's topological proof that every written alphabet is... by lzralbu on Wednesday July 24, 2024 @11:55PM (#41064559)

Note that $A_k = \left[\frac{1}{k}, \frac{2}{k}\right] \times \left[\frac{1}{k}, \frac{2}{k}\right] \to \{(0, 0)\}$ in $[0,1]\times [0, 1]$ with the Hausdoff metric, so it is a Cauchy sequence in $X = (0,1) \times (0, 1)$ but without convergent subsequence, i.e., $F(X)$ is not complete, hence not compact so the finite subcover argument fails.

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