actually a very good exercise to understand why this argument doesn’t work. you’ll learn something about the nature of continuity and what it really means to repeat a process infinitely many times
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okay i'm gonna be real
i have a math degree but don't really understand why this is the case. perhaps a poor reflection on me/my school
i get that while the distances between the curves goes to 0, that doesn't mean the curve lengths do, but just saying that feels handwavy
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so, less handwavily: what this argument shows is that any topology on the space of curves must have at least one of two properties. either the sequence of PL curves fails fo converge to the circle, or length fails to be continuous wrt the topology
If I the subset of closed loops containing loops formed by a finite number of horizontal and vertical lines, and make a distance on it as the area of the symmetric difference of two enclosed shapes, the resulting space should have circles in its closure...
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... and if I define the perimeter on the closure to be the limit of the perimeters of loops converging to that shape, then unit circles have perimeter 4. I still don't really see where the issue arises
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length isn't continuous, that's about it. not very surprising.
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