On the topological aspects of the theory of represented spaces

Article type: Research Article

Authors: Pauly, Arno^{; *}

Affiliations: Computer Laboratory, University of Cambridge, Cambridge, UK. Arno.Pauly@cl.cam.ac.uk. www.cl.cam.ac.uk/~amp66/

Note: [*] The author has moved to the Département d’Informatique, Université libre de Bruxelles, Belgium.

Abstract: Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by Escardó, Schröder, and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and – closely linked to these – properties of represented spaces such as compactness, overtness and separation principles. Both the derived spaces and the properties are introduced by demanding the computability of certain mappings, and it is demonstrated that typically various interesting mappings induce the same property.

Keywords: Computable analysis, realizability, synthetic topology, admissible representation

DOI: 10.3233/COM-150049

Journal: Computability, vol. 5, no. 2, pp. 159-180, 2016