Hello,
We know that A is Wadge reducible to B if there is a continuous map $f$ such that A is the preimage of B via $f$, and the Wadge order is defined by $A\leq_{w}B$ if $A$ is Wagde reducible to $B$ or $A$ is Wadge reducible to $B^c$. Martin proved this is a prewellordering on sets of reals. The Wadge rank of $A$ is the order-type of $A$ in this prewellordering.
My question is if $A\equiv_{w}B$, what can we know about the Wadge rank of $A\cup B$? The union can be weaker than $A$ and can be stronger than $A$. But is there some bound? For example, is it true that the Wadge rank of $A\cup B$ is smaller than (the Wadge rank of $A)+\omega_1$? And how about a countable union of sets with the same Wadge rank?
Thank you!
