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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Removing an infinite amount of corners results in smth that ismt a circle, at a really close up view there would be a jagged staircase next to a smooth round circle. Not equal
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Anything you can put together with only horizontal and vertical lines is not even in the same conceptual space as a circle 🤷♂️ I don't make the rules
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Sooo... Topology = conceptual space?
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Replying to @architectonyx
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
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if you've taken calculus the easiest way of saying it is that a topology on a set (here, the set of all curves, or maybe all "reasonable curves") is a way of assigning limits to sequences in that set; this isn't quite true but it's good enough to start with
like if you want to make precise the sense in which the sequence of really jagged curves "converges to," or doesn't converge to, the circle, the way you do that in modern mathematics is to put a topology on the set of curves
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