The Wadge Hierarchy on N^N endowed with the co- compact topology: the ∑^0_2-degrees

This thesis aims at investigating the Wadge hierarchy on other T1, non-T2 spaces X. In so doing, a first step could be to consider uncountable spaces X endowed with the co-compact topology, for a fixed topology on X. Everything in Descriptive Set Theory gets easier when dealing with the Baire space or its subspaces, so this thesis will analize the structure of the Wadge hierarchy on (N^N, \tau) where the open sets of \tau are the ones whose complement is compact in the product topology of N^N. This topology is very wild (for example, it is not first-countable, hyperconnected and sequentially compact), however a segment of the hierarchy will be fully revealed, namely the degrees represented by the sets that are \Sigma^0_2 in the product topology: although this preorder differs a lot from N^N W , its computing is greatly linked with the already known Wadge order on the Cantor Space2^N and nevertheless it presents some odd properties.