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A few years ago I saw a reference to two articles by two prominent 19th Century mathematicians, whose names I can no longer remember, to constructions of bicontinuous bijections between an interval, maybe open, maybe closed, of the reals, and a square, similarly open or closed, in 2-dim. Euclidean space, which I wouldn't have believed possible. However, I couldn't find any access to the articles. Can anyone provide me with a description of such a construction, or a way to access such an article?

On the other hand, can anyone provide me with a proof, or a way to access a proof, that such a construction is impossible?

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  • $\begingroup$ Do you mean this? $\endgroup$ – poyea Mar 14 at 8:07
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No such thing is possible. if you remove a point in the middle of the interval it becomes disconnected. If you remove a point from a square it remains connected. So no such function can exist.

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