COUNTEREXAMPLES IN EFFECTIVE TOPOLOGY

Abstract We prove that there exists a left-c.e. Polish space not homeomorphic to any right-c.e. space. Combined with some other recent works (to be cited), this finishes the task of comparing all classical notions of effective presentability of Polish spaces that frequently occur in the literature up to homeomorphism . We employ our techniques to provide a new, relatively straightforward construction of a computable Polish space K not homeomorphic to any computably compact space. We also show that the Banach space $C(K;\mathbb {R})$ has a computable Banach copy; this gives a negative answer to a question raised by McNicholl. We also give an example of a space that has both a left-c.e. and a right-c.e. presentation, yet it is not homeomorphic to any computable Polish space. In addition, we provide an example of a $\Delta ^0_2$ Polish space that lacks both a left-c.e. and a right-c.e. copy, up to homeomorphism.