Currently, I am struggling to fully understand the following theorem from Alexander Kechris' book “Classical Descriptive Set Theory”:
The last sentence is unclear to me. Why is the homeomorphic copy of $\mathcal{C}$ with respect to $\mathcal{T}_A$ also homeomorphic with respect to $\mathcal{T}$?
It has been shown that there is $Y \subseteq A$ s.t. $(Y, \mathcal{T}_A|Y)$ is homeomorphic to $\mathcal{C}$. But why is $(Y, \mathcal{T}|Y)$ also homeomorphic to $\mathcal{C}$?
A Cantor set is compact, so the identity $(C,\mathcal{T}_A) \to (C,\mathcal{T})$ is a homeomorphism. IT follows that if $(Y, \mathcal{T_A}|Y)$ is homeomorphic to the Cantor set, then so is $(Y, \mathcal{T}|Y)$.