How is it that there are 'gaps' in rational numbers and yet between any two rational numbers, there exists another rational number?

Question

If there are gaps in rational numbers then lets assume we have a gap between a and b, both being rational. Then we have $\frac{a+b}{2}$ which is inside the gap which essentially makes it a non-gap. What am I getting wrong?

Answer 1

A gap in $\Bbb Q$ means there exist non-empty sets $A, B$ with $\Bbb Q=A\cup B,$ such that (i) every $a\in A$ is less than every $b\in B,$ and (ii) $A$ has no largest member and $B$ has no smallest member. It does NOT mean that there are rationals $x, y$ with $x<y$ such that no rational is between $x$ and $y$. No such $x,y$ exist but if they did, the pair $(x,y)$ would be called a jump.

Example: No $x\in \Bbb Q$ satisfies $x^2=2.$ Let $B=\{b\in \Bbb Q: 0<b\land b^2>2\}$ and let $A=\Bbb Q \setminus B.$ If $b\in B$ then $b>\frac {1}{2}(b+\frac {2}{b})\in B,$ so $B$ has no least member. This implies that $\{a\in A:a>0\}=\{\frac {2}{b}: b\in B\}$ has no largest member. So $A$ has no largest member.