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Let $X$ be a real separable Banach space. Let $A\subset X$ be the enumerable set, given by the separability. How can i define a continuous "bijective" function $f:A\rightarrow\mathbb{Q}$, where im assuming the induced topology in both space (in case $\mathbb{Q}\subset\mathbb{R}$).

Thanks for the help.

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    What properties do you want $f$ to have? I'm assuming you want more than just continuous, since otherwise any constant function will work.2012-10-20
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    @froggie, you are right, i forgot they key word. Im gonna edit2012-10-20
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    This is an old theorem of Sierpinski. In fact, any countable metric space without isolated points is homeomorphic to $\mathbb{Q}$.2012-10-20
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    interesting, do you have any reference? or maybe the proof?2012-10-20
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    Here's a [reference](http://at.yorku.ca/p/a/c/a/25.htm). I worked through an argument once years ago. If I can reconstruct it, I'll post an answer, but it's more than just a few lines.2012-10-20

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