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Topic: Intersecting Images (Read 1322 times) |
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ThudnBlunder
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Intersecting Images
« on: Mar 5th, 2005, 10:43am » |
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Find a function R -> R such that the image of any interval intersects with every other interval.
Can such a function have an inverse?
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Obob
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Re: Intersecting Images
« Reply #1 on: Mar 5th, 2005, 1:58pm » |
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Use the Axiom of Choice to find a set of coset representatives for the abelain group R/Q. This group is uncountably infinite (since Q is countable and R is uncountable), so there exists a bijection between the set of coset representatives and R, call this function g. Define a function f:R->R by declaring f(x)=g(x+Q); that is, f evaluated at x is equal to g evaluated at the unique coset representative which differes from x only by an element of Q. Then f is a well-defined function, and it has the property that the image of any interval is all of R since every interval contains a complete set of coset representatives for R/Q!
I haven't been able to figure out the inverse question, I'll leave that one for now.
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ThudnBlunder
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Re: Intersecting Images
« Reply #2 on: Mar 12th, 2005, 11:23pm » |
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Any ideas on a function constructibe in ZF?
Is it bijective?
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| « Last Edit: Mar 13th, 2005, 1:35am by ThudnBlunder » |
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Obob
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Re: Intersecting Images
« Reply #3 on: Mar 13th, 2005, 11:17am » |
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This function is obviously not bijective. While its certainly possible that there is some such function which is constructible in ZF, it seems like it would be extremely difficult to construct.
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