It is well-known that [i.e., I think it's true but can't be bothered to look up the reference] the pre-image of an open set under a continuous function is also open [note that it doesn't work in the opposite direction!]. See any elementary text on point-set topology. So your "nice" function does not exist. As to a succinct definition of a bijection, nothing suggests itself right now ... Anybody else? WFL On 11/30/06, Kerry Mitchell <lkmitch@gmail.com> wrote:
More of an intellectual curiosity question than anything useful. Is there a "nice" function that is one-to-one and continuous that maps the closed interval [0, 1] to the open interval (0, 1)? Ideally, such a "nice" function would be not a piecewise function or a power series and something that could be implemented with standard library functions. If such a thing doesn't exist, is there at least a "not nice" function that is one-to-one and maps the intervals? If not, why not?
Thanks for any pointers or wisdom, Kerry Mitchell -- lkmitch@gmail.com www.fractalus.com/kerry
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