# Order preserving injection $f$ from set of rationals $Q$ into $R$ with discrete image.

How to construct an order-preserving injection $$f:Q\rightarrow R$$ , such that the image of $$f$$ is discrete subspace of $$R$$ (set of reals).

Enumerate $$Q = \{q_n : n \in \omega \}$$. We are going to define $$f(q_n)$$ by induction on $$n$$. This is done by defining an auxiliary sequence $$(I_n)_{n \in \omega}$$ of disjoint open intervals which do not overlap and have distinct endpoints so that $$q_n \in I_n$$ for every $$n$$.

Start by mapping $$q_0$$ anywhere you want and take $$I_0$$ to be any bounded open interval around $$f(q_0)$$.

Having defined $$(f(q_i))_{i, map $$q_n$$ aproprietly so that $$f(q_n) \notin \bigcup_{i. Now we can still find an open interval $$I_n$$ around $$f(q_n)$$ which is disjoint from $$\bigcup_{i and does not share any endpoints.

I leave the details to you.

• How is such $f$ order preserving? And do I have to specify these disjoint open intervals, say from Cantors middle third set?
– ersh
May 7, 2019 at 18:54
• This is what I meant with "map $q_n$ approprietly". $q_n$ has a certain position within $\{q_i : i \leq n \}$, say for example $q_j < q_n < q_i$ with $j,i <n$ where $q_j$ is chosen largest with this property and $q_i$ smallest. Then we would map $q_n$ to something in the intervall $(f(q_j), f(q_i))$. Let $a$ be the right endpoint of $I_j$ and $b$ the left endpoint of $I_i$, then $q_j < a < b < q_i$. So we can actually chose $f(q_n)$ between $a$ and $b$. Then let $I_n = (c,d)$ so that $a<c<d<b$ and $f(q_n) \in (c,d)$. May 7, 2019 at 19:17
• Any choice of $(c,d)$ is ok, e.g. $c = f(q_n) - \frac{f(q_n) -a}{2}$, $d = f(q_n) + \frac{b -f(q_n)}{2}$. May 7, 2019 at 19:17
• You can also see that the set $\{ \sum_{i < \vert s \vert } (-1)^{1-s(i)}3^{-i} : s \in 2^{< \mathbb{N}} \}$ is countable dense in itself and discrete. Maybe that's the example you were looking for (or asked for), since you are talking about the Cantor set. May 7, 2019 at 19:30
• @ersh Yes the construction works for any countable linear order. May 7, 2019 at 21:22