I was troubled by the definition of cross ratio,give four ordered points $X,A,B,Y$ the cross ratio should be $\frac{XB}{XY}/\frac{AB}{AY}$.But according to the bove definition in the screenshot,the cross ratio is $\frac{XB}{XA}\cdot \frac{AY}{BY}$. Where is wrong?
Don't mix the order of symbols in the formula and the order of points along the line.
The points are in order $X,A,B,Y$ along the line, because $A,B$ are inside $\Omega$ while $X,Y$ lie on the boundary of $\Omega$. On the other hand, the order in the cross ratio formula is a different, if you follow typical conventions:
$$d(A,B)=\log\left\lvert\operatorname{CR}(Y,X;A,B)\right\rvert=\log\left( \frac{\lvert YA\rvert\cdot\lvert XB\rvert}{\lvert YB\rvert\cdot\lvert XA\rvert}\right)$$
Let's look at a few special cases:
So if you have the order as stated, with $B$ lying between $A$ and $Y$, then you get a cross ratio between $1$ and $\infty$ and a distance between $0$ and $\infty$ i.e. a positive distance. If on the other hand you were to change the order, so that $B$ lies between $X$ and $A$, then you'd get a logarithm less than $1$ and a negative distance. That's why the order of points along the line was specified the way it was.
Advanced use cases: For $B$ outside $\Omega$, i.e. order $X,A,Y,B$ you'd get a negative cross ratio, and a purely imaginary distance. (Writing the cross ratio using absolute value bars would be fairly confusing in this setup.) In some contexts this view might still make sense, to provide a more complete picture. In those situations you'd want to keep in mind that the imaginary part of a distance is no longer uniquely defined, since any multiple of $2\pi$ can be added to it to express the same cross ratio.