Borel space of dynamical subsystems

Question

Let $ T $ be an invertible function acting by homeomorphisms on a compact Polish space $ X $ (i.e., a continuous action of $ \mathbb{Z} $). Denote by $ [x] $ the orbit of $ x $ under $ T $: $ [x] = \{\dots, T^{-1}x, x, Tx, \dots \} $. Recall $ K(X) $ is the space of compact subsets of $ X $ with the Hausdroff metric (i.e., Vietoris topology). What is a nice way to show formally that $ x \mapsto \overline{[x]} $ is a Borel map?

Answer 1

Define $ f_n : X \to K(X) $ by $f_n(x) = \{ T^{-n}x, \dots, x, \dots, T^nx \} $. As these are maps between metrics spaces, to see that each $ f_n $ is continuous it suffices to check that $ f_n(x_k) \to_{k \to \infty} f_n(x) $, which is clear. Notice that $ f $ is the pointwise-limit of $ f_n $. By Theorem 1.10 of https://webusers.imj-prg.fr/~dominique.lecomte/Chapitres/5-Borel%20sets%20and%20functions.pdf (specifically using the fact that the co-domain is separable), $ f $ is Borel.