elementary set theory - Bijective map from $Bbb Z$ to $Bbb Q$

There exists a map $f: \Bbb Z\rightarrow \Bbb Q $ such that $f$ is

A. Bijective and increasing

B. Onto and decreasing

C. Bijective and satisfies $f(n)\ge 0$ if $n\le 0$

D. Has uncountable images

Now option D. is absurd . Option C. is given to be the correct answer.I was thinking since both sets are countable bijection is obvious. Now why cannot be increasing ? I could map $0$ to $0$ and the negative integers to the negative rationals and positive integers to the positive rationals. And if increasing would be possible just interchanging signs would give the decreasing map. So none is possible but why?

Answer

The problem is that, while the cardinality of $\mathbb{Z}$ and $\mathbb{Q}$ is the same, the topology is different. Consider the following: Given two integers $a$ and $b$ with $a < b$, can we say how many integers $c$ satisfy $a < c < b$? Now ask the same question for the rational numbers.

This leads to problems. For example, lets say $a$ and $b$ are two such integers with $k$ other integers between them. And we can even assume the $f(a) = r_{1} < r_{2} = f(b)$. But now we can find an increasing sequence of$k+1$ rational numbers (at least) between $r_{1}$ and $r_{2}$, and only $k$ potential integers to use as their preimages.