Let $X$ be a metric space. Suppose $(f_{\eta})_{\eta < \omega_1}$ is a sequence of real-valued functions on $X$ where $\omega_1$ is the first uncountable ordinal.
Let's say we have a sequence of subsets $(A_{\xi})_{\xi < \omega_1}$ such that $A_{\eta}$ is strictly decreasing, that is, $A_{\eta+1} \subsetneq A_{\eta} $ all $\eta < \omega_1$. Furthermore, $A_1$ is related to $f_1$, $A_2$ is related to $f_3$, $A_3$ is related to $f_5$, $A_4$ is related to $f_7$, and so forth.
When both $\xi$ and $\eta$ exceed $\omega$, then we relate them in the following way: $A_{\omega+1}$ is related to $f_{\omega +1}$, $A_{\omega +2}$ is related to $f_{\omega +3}$, $A_{\omega +3}$ is related to $f_{\omega+5}$, $A_{\omega +4}$ is related to $f_{\omega+7}$, and so forth.
When both $\eta$ and $\xi$ exceed $\omega \cdot 2$, then we relate them in the following way: $A_{\omega \cdot 2+1}$ is related to $f_{\omega \cdot 2 +1}$, $A_{\omega \cdot 2 +2}$ is related to $f_{\omega \cdot 2 +3}$, $A_{\omega \cdot 2 +3}$ is related to $f_{\omega \cdot 2+5}$, $A_{\omega \cdot 2 +4}$ is related to $f_{\omega \cdot 2+7}$, and so forth.
Question: How to obtain a general formula so that we have $\xi$ in terms of $\eta$? If $\xi$ and $\eta$ are just natural numbers, that is, before they exceed $\omega$, one can have $\xi=\frac{\eta +1}{2}$.
However, division is not well-defined for ordinals. Can someone give some hint on how to obtain a general formula?
UPDATE: Is the following statement correct?
If $\eta = \omega^{\alpha} \cdot \beta + \gamma$ for some $\alpha, \beta$ and $\gamma$, then $\xi = \omega^\alpha \cdot \beta + \frac{\gamma +1}{2}$.
You have specified a relationship between the $\xi$s and the $\eta$s over the naturals. You have not specified what happens at limits or beyond. One reasonable approach if the $f$s and $A$s are convergent would be to say $f_\omega$ is related to $A_\omega$ and then keep going. In that case the infinite part of the ordinal would be the same between $f$ and $A$ and you would apply the same function to the finite part.
With the edit, you are requesting exactly what I suggested above. Write the indices in Cantor normal form, copy over all the terms involving $\omega$, and apply your function $\xi=\frac{\eta +1}{2}$ to the finite part. What you added in UPDATE is correct, but does not cover all the cases. You can have $\eta=\omega^3\cdot 7+ \omega^2\cdot 9 + 1007$ and would want $\xi=\omega^3\cdot 7+ \omega^2\cdot 9 + \frac {1007+1}2$.