## Damiani Fernando

PhD Graduate**PhD program:**: XXXIII

**supervisor**: E. Rogora

**advisor**: R. Camerlo

**Thesis title:**The Wadge hierarchy on $\mathbb N^\mathbb N$ endowed with the co-compact topology: the $\Sigma^0_2$-degrees.

The Wadge hierarchy on topological spaces different from the Baire space has been subject of intensive study in the last decade and much has been discovered for non-zero dimensional Polish spaces. For non-metrizable spaces, the problem is very open and the hierarchy's behaviour can be very wild, for instance, with the presence of arbitrarily large finite antichains or even antichains of cardinality the continuum, as Camerlo showed in the context of affine varieties. The aim of this thesis is to start a study of the Wadge hierarchy on a specific non-metrizable space: the set $\mathbb N^\mathbb N$ endowed with the co-compact topology generated by the Baire space: the open sets are those with compact complement in the usual product topology of $\mathbb N^\mathbb N$. It turns out that the continuous reducibility between two "very small" subsets, namely those sets contained in a compact subset of the Baire space, is related to that between the dense subsets of the Cantor space. Thus an analisys of the degrees of the Wadge hierarchy on $2^\mathbb N$ is carried out, paying attention on those degrees containing a dense subsets of the Cantor space. For what concern the non-very small sets, we have to proceed manually: this work highlights the Wadge hierarchy restricted to the the class of those sets that are $\Sigma^0_2$ in the Baire space (and then also to the dual class $\Pi^0_2$). Antichains of length 4 come up and, starting from the top level of this segment of the hierarchy, that is the degree containing all clopen sets of the Baire space, there is an increasing sequence of degrees obtained with operations similar to those by Wadge. Actually, some degrees isolated in the thesis contain also some $\Sigma^0_3$ set and this lets conjecture that all sets that are Borel in the co-compact topoly reduce to the clopen sets.