A recent area of interest in computable topology compares different notions of effective presentability for topological spaces. In this paper, we show that up to isometry, there is a compact connected Polish space that has both left-c.e. and right-c.e. Polish presentations, but has no computable Polish presentation. We also construct a Polish group that has both left-c.e. and right-c.e. Polish group presentations, but lacks a computable Polish presentation, up to topological isomorphism.

Abstract We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions based on classical notions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare our results with similar results in computable topology.

This paper contributes to a systematic study of punctual structures , which are structures computable without delay. The (punctual) degree structure induced by treating “being primitive recursively isomorphic” as a reduction provides insight into the different speeds of enumerations of a given structure. In this paper, we work towards a classification of density of the punctual degrees for linear orders. More specifically, we construct a discrete linear order whose punctual degrees are not dense.