How about the Wadge rank of A union B when A and B are Wadge equivalent? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:22:05Z http://mathoverflow.net/feeds/question/100595 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100595/how-about-the-wadge-rank-of-a-union-b-when-a-and-b-are-wadge-equivalent How about the Wadge rank of A union B when A and B are Wadge equivalent? Grace 2012-06-25T13:59:18Z 2012-06-25T13:59:18Z <p>Hello,</p> <p>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.</p> <p>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?</p> <p>Thank you!</p>