1
$\begingroup$

In this post, we are not considering $\mathbb{R}$, we only focus on the rationals.
To prove $E=\{q : q\in \mathbb{Q} \wedge 20$ for every such $x$ such that for every $q \in N_r(x)=\{q : q\in \mathbb{Q} \wedge |x-q| What I do is say that $(x+r)^2<2 \Rightarrow r<\sqrt{2}-x$ but the solutions say to set $r=\min (\sqrt{\frac{2-x^2}{3}},\frac{2-x^2}{3|x|})$ and while I can prove that such an $r$ gives the desired neighborhood, I do not know how I would have come up with such an expression. How could I obtain such an expression on my own?

  • 1
    As a subspace of the usual topology on $\mathbb R$, $E$ is not closed since it contains a sequence of rationals that can approximate $\sqrt 3$ (which does not belong to $E$) to an arbitrary precision. – 2017-01-07
  • 0
    @Adriano but we are not considering $\mathbb{R}$, we are focusing on the rationals only – 2017-01-07
  • 0
    @Crostul I added that information – 2017-01-07
  • 0
    "Closed" is something you can ask for a ___subset___ of a metric space. If the subset is not proper, i.e. if the subset is the entire space, the question is not interesting because the entire space is always closed ___in___ itself. So the question "is $E$ closed in (the metric space) $E$?" is not interesting; its answer is "yes". The question "is $E$ closed in $\mathbb{R}$?" clearly has answer "no". But the question you are trying to formulate is actually "is $E$ closed in $\mathbb{Q}$?" – 2017-01-07
  • 0
    You could change the title into ___Prove that the subset $\{q : q\in \mathbb{Q} \wedge 2$\mathbb{Q}$___ – 2017-01-07

1 Answers 1

1

If $q^2\ge 3,$ then $q>\sqrt{3}$ or $q<-\sqrt{3}$ (since $\pm\sqrt{3}\notin\mathbb{Q}$). So if $q>\sqrt{3}$ take an open ball of all rationals less than $ r= q-\sqrt{3}$ distance from $q$ and this will be a neighborhood of $q$ that is contained in the complement. (Similar for $q <-\sqrt{3}.)$

If $q^2 \le 2$ then $-\sqrt{2}

If this (or something like it) was your solution, I see no problem. I have no idea where that expression in your class solution comes from.

  • 0
    How could I have derived those radii more formally? I can intuitively derive them, but I cannit make them fall out of some equation or inequality. – 2017-01-07