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I am having a problem understanding what's wrong with this:I know statement 1 is false and 2 is true but I can't find the mistake. 1)Any open set in $R$ is a countable union of disjoint open intervals. So if we take any open set containing all rationals its complement contains only irrational points. But because endpoint of one interval is the starting point of the next, we can associate the right endpoint of any interval(which is irrational) to each interval. Hence the number of irrational points in the complement is countable. 2)We can construct an open set in $R$ with measure less than $1$ containing all rationals. Hence its complement will have infinite measure and is uncountable.

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  • $\begingroup$ Presumably by "measure less than $1$" you mean "finite measure," since $\mathbb{R}$ has measure $\infty$, not $1$ ... $\endgroup$ Commented Oct 23, 2017 at 5:30
  • $\begingroup$ I was talking about the open set containing all rationals to have measure less than 1 $\endgroup$
    – CoffeeCCD
    Commented Oct 23, 2017 at 5:34

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Actually, both (1) and (2) are correct! The error is when you say:

Hence the number of irrational points in the complement is countable.

This only works if you know that every irrational is the right endpoint of some interval in the collection. And this is not true.

For example, consider the set of open intervals $$\{(0, 3), (3, 3.1), (3.1, 3.14), (3.14, 3.141), ..., (3.142, 3.15), (3.15,3.2), (3.2, 4), (4, 5)\}.$$ It should be clear that these intervals "converge to $\pi$" - so $\pi$ isn't the right endpoint of any one of these intervals.

As counterintuitive as it may seem, in an open cover of the rationals with finite measure, most irrationals are not the endpoints of any of the intervals in the cover.

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  • $\begingroup$ So I am missing out on some irrational points in the complement in a case such as this. I got it. Thanks, Yeah I meant what is wrong with my statement of the complement being countable. I know the characterisation of open sets was correct. $\endgroup$
    – CoffeeCCD
    Commented Oct 23, 2017 at 5:33

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