Is there a bijection of the plane sending every circle to a square?

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# Squaring the circle

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Is there a bijection of the plane sending every circle to a square?

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Consider an infinite family of circles such that their pairwise intersection is a single point , as in the construction of the hawaiian earring. If such a bijection existed, it would have to send this family to an infinite family of squares that pairwise intersect in a single point, which is clearly impossible.

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