We prove that the B\"uchi topology and the automatic topology are Polish. We also show that this cannot be fully extended to the case of a space of infinite labelled binary trees; in particular the B\"uchi and the Muller topologies are not Polish in this case.Comment: This paper is an extended version of a paper which appeared in the proceedings of the 26th EACSL Annual Conference on Computer Science and Logic, CSL 2017. The main addition with regard to the conference paper consists in the study of the B\"uchi topology and of the Muller topology in the case of a space of trees, which now forms Section 4

@article{paperbot797,
  title = {Polishness of some topologies related to word or tree automata},
  author = {Olivier Finkel and Olivier Carton and Dominique Lecomte},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 15, Issue 2},
  year = {2019},
  doi = {10.23638/lmcs-15(2:9)2019}
}