Year: 2025
Type: article
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.
Source: Journal of Symbolic Logic
DOI: 10.1017/jsl.2025.25
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W4408801945