Computable metrics above the standard real metric

R. A. Kornev

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 1, pp. 377-392

Résumé

We construct a sequence of computable real metrics pairwise incomparable under weak reducibility $\leq_{ch}$ and located above the standard real metric w. r. t. computable reducibility $\leq_c$. Iterating the construction, we obtain that the ordering $(P(\omega),\subseteq)$ of subsets of $\omega$ is embeddable into the ordering of $ch$-degrees of real metrics above the standard metric. It is also proved that the countable atomless Boolean algebra is embeddable with preservation of joins and meets into the ordering of $c$-degrees of computable real metrics.

Détail

Keywords: computable metric space, representation of real numbers, Cauchy representation, reducibility of representations, computable analysis.