Geodesic
Reducibility of computable metrics on the real line
Algebra i logika, Tome 56 (2017) no. 4, pp. 453-476

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We study computable reducibility of computable metrics on $\mathbf R$ induced by reducibility of their respective Cauchy representations. It is proved that this ordering has a subordering isomorphic to an arbitrary countable tree. Also we introduce a weak version of computable reducibility and construct a countable antichain of computable metrics that are incomparable with respect to it. Informally, copies of the real line equipped with these metrics are pairwise homeomorphic but not computably homeomorphic.
Keywords: computable metric space, Cauchy representation, reducibility of representations. 
@article{AL_2017_56_4_a4,
     author = {R. A. Kornev},
     title = {Reducibility of computable metrics on the real line},
     journal = {Algebra i logika},
     pages = {453--476},
     publisher = {mathdoc},
     volume = {56},
     number = {4},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2017_56_4_a4/}
}
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R. A. Kornev. Reducibility of computable metrics on the real line. Algebra i logika, Tome 56 (2017) no. 4, pp. 453-476. http://geodesic.mathdoc.fr/item/AL_2017_56_4_a4/