# Closedness of the subset [0,1] in $\Bbb{Q}$

I'm working through the definitions given in 2.18 in Baby Rudin, and I was wondering how to prove that $$E = [0,1] \cap \Bbb{Q}$$ is closed in $$\Bbb{Q}$$.

Clearly, the set $$[0,1] \cap \Bbb{R}$$ is closed in $$\Bbb{R}$$, since for all neighborhoods of a point $$p \in [0,1]$$, we can always find a $$q \neq p$$ such that $$q \in [0,1]$$. Intuitively, this is because for all $$p \in [0,1] \cap \Bbb{R}$$, we can always find a $$q \in [0,1] \cap \Bbb{R}$$ such that $$p - \epsilon < q < p$$ or $$p < q < p+\epsilon$$ for all $$\epsilon >0$$.

However, that the set $$E = [0,1] \cap \Bbb{Q}$$ is closed in $$\Bbb{Q}$$ is sort of clear, but not intuitive to me, since $$\Bbb{Q}$$ is not uncountably infinite as $$\Bbb{R}$$ is. Does one have to use that fact that it is closed in $$\Bbb{R}$$ and that $$\Bbb{Q}$$ is dense in $$\Bbb{R}$$? How would one solve it from first principles? i.e. Something along the lines of: Consider an arbitrary $$p \in [0,1] \cap \Bbb{Q}$$, we show that it is a limit point by...

Any help would be greatly appreciated!

• Since $[0,1]\not\subset\mathbb Q$, your question makes no sense. Did you mean that $[0,1]\cap\mathbb{Q}$ is a closed subset of $\mathbb Q$? And you do you mean when you say that $\mathbb Q$ is not continuous? Sep 11, 2018 at 13:47
• "$\Bbb Q$ is not continuous"? It is functions that may or may not be continuous, not sets. Just show the complement of $[0,1]\cap\Bbb Q$ is open in $\Bbb Q$. Sep 11, 2018 at 13:48
• @JoséCarlosSantos Interval notation works in any total order. It's most commonly used in $\Bbb R$, but I've seen it used generally as well. Sep 11, 2018 at 13:52
• @Arthur I am not sure that that's clear in the OP's mind. Sep 11, 2018 at 13:53
• Then you should type $[0,1]\cap \mathbb Q$.
– xbh
Sep 11, 2018 at 13:57

I'm sorry to say I couldn't follow your argument that it is intuitive $[0,1]$ is closed in $\mathbb R$ at all.

$[0,1]$ is closed (in $\mathbb R$) because all its limit points are in in $[0,1]$. That is; If $p \in \mathbb R$ and for every $\epsilon > 0$ it will always be that the interval $(p-\epsilon, p+\epsilon)$ will contain a point $q \in [0,1]$, then we can conclude that $p$, itself, is in $[0,1]$.

Pf: If $p < 0$ and $\epsilon = 0 - p= |p|$ then $(p-\epsilon, p + \epsilon) = (2p, 0)$ is disjoint from $[0,1]$ and there is no point, $q$, in common with $[0,1]$. And if $p > 0$ and $\epsilon = p - 1$ then $(p-\epsilon, p+ \epsilon) = (1, 2p - 1)$ and there is no point, $q$, in common with $[0,1]$.

So if $p$ is a limit point (if there are any limit points, there might not be) then it would have to be that $0 \le p \le 1$ so $p \in [0,1]$.

Thus by definition $[0,1]$ is closed.

.....

But notice there is nothing in that proof that pertains to any property of $\mathbb R$ and it would have held true for any subspace of $\mathbb R$ at all. Everything we said would be true for $\mathbb Q$ or any $B \subset \mathbb R$.

Also, although the proof to show $[0,1]$ was closed was particular to $[0,1]$, there is no proof of any other set being closed that would rely upon the space.

Claim: If $A$ is closed in space $X$ then $A\cap Y$ is closed in $Y \subset X$.

Pf: Let $p \in Y$ be a limit point of $A\cap Y$ then every neighborhood, $N$ of $p$ contains a point $q; q \ne p; q \in A\cap Y\subset A$. But then $N$ is a subset of a neighborhood $N'\subset X$. $q \in A$ and $q \in N'$. So $p$ is a limit point of $A$. And as $A$ is closed in $X$ that means $p \in A$. And as $p \in Y$. So $p \in A\cup Y$. So $A\cup Y$ is closed in $Y$.

The density of $\mathbb Q$ plays no role here. If $A$ and $B$ are subsets of $\mathbb R$ and $A$ is a closed subset of $\mathbb R$, then $A\cap B$ is a closed subset of $B$. That's so because the complement of $A\cap B$ in $B$ is equal to $A^\complement\cap B$. Now, since $A$ is a closed subset of $\mathbb R$, $A^\complement$ is an open subset of $\mathbb R$ and therefore $A^\complement\cap B$ is an open subset of $B$.

• This was helpful, and I now see why it doesn't matter that $\Bbb{Q}$ is dense in the reals. But in particular what I wanted was a way to show that an arbitrary $p \in [0,1] \cap \Bbb{Q}$ is a limit point in $[0,1] \cap \Bbb{Q}$ Sep 11, 2018 at 14:06
• @PhysMath Why? The set $(0,1)$ is not closed in $\mathbb R$. Nevertheless, each of its points is a limit point of $(0,1)$. Sep 11, 2018 at 14:08
• @PhysMath You got the definition wrong. The direction was reversed. Your goal should be that every limit point of $[0,1] \cap \mathbb Q$ lies in $[0,1] \cap \mathbb Q$, not the other direction.
– xbh
Sep 11, 2018 at 14:19

A bit formal :

Baby Rudin: Theorem 2.30.

Suppose $Y \subset X$. $A$ subset $E$ of $Y$ is open relative to $Y$

$\iff$

$E=Y \cap G$ for some open subset $G$ of $X$.

$B= [0,1]$ is closed in $\mathbb{R}$.

$B^c:= \mathbb{R}$\ $[0,1]$ is open in $\mathbb{R}$.

Hence

$(B^c \cap \mathbb{Q})$ is open in $\mathbb{Q}$.

The complement of $(B^c \cap \mathbb{Q})$ in $\mathbb{Q}$ is closed in $\mathbb{Q}$ :

$\mathbb{Q} - (B^c \cap \mathbb{Q})=$

$\mathbb{Q} \cap (B^c \cap \mathbb{Q})^c =$

$\mathbb{Q} \cap (B \cup \mathbb{Q}^c)=$

$(\mathbb{Q} \cap B)\cup (\mathbb{Q} \cap \mathbb{Q}^c)=$

$\mathbb{Q} \cap B$.

• Absolutely true. Theorem 2.30 basically says this must be true. The question is intuitively why is 2.30 true... and I guess.... well, whatever we say about points being in or not in neighborhoods or a set E will be true of points being in or not in intersections of neighborhoods or a set E with a subspace of X. Once we intuitively "grok" the definitions the result is obvious. (The only issue of doubt a student could have, I think, is there must be some intuitive real space "out there" that border of the subspace "cuts off". In which case we need to understand what a subspace means.) Sep 11, 2018 at 18:34
• Thank you for your comment.Your answer and comment show that you are well at ease, while I am at times happy to get from A to B (prove something for a closed set B, starting with 2.30 for an open set A :))Greetings. Sep 11, 2018 at 20:05