912 research outputs found
Effectivity on Continuous Functions in Topological Spaces
AbstractIn this paper we investigate aspects of effectivity and computability on partial continuous functions in topological spaces. We use the framework of TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notations and representations. We generalize the representations introduced in [Weihrauch, K., “Computable Analysis,” Springer, Berlin, 2000] for the Euclidean case to computable T0-spaces and computably locally compact Hausdorff spaces respectively. We show their equivalence and in particular, prove an effective version of the Stone-Weierstrass approximation theorem
Products of effective topological spaces and a uniformly computable Tychonoff Theorem
This article is a fundamental study in computable analysis. In the framework
of Type-2 effectivity, TTE, we investigate computability aspects on finite and
infinite products of effective topological spaces. For obtaining uniform
results we introduce natural multi-representations of the class of all
effective topological spaces, of their points, of their subsets and of their
compact subsets. We show that the binary, finite and countable product
operations on effective topological spaces are computable. For spaces with
non-empty base sets the factors can be retrieved from the products. We study
computability of the product operations on points, on arbitrary subsets and on
compact subsets. For the case of compact sets the results are uniformly
computable versions of Tychonoff's Theorem (stating that every Cartesian
product of compact spaces is compact) for both, the cover multi-representation
and the "minimal cover" multi-representation
A rich hierarchy of functionals of finite types
We are considering typed hierarchies of total, continuous functionals using
complete, separable metric spaces at the base types. We pay special attention
to the so called Urysohn space constructed by P. Urysohn. One of the properties
of the Urysohn space is that every other separable metric space can be
isometrically embedded into it. We discuss why the Urysohn space may be
considered as the universal model of possibly infinitary outputs of algorithms.
The main result is that all our typed hierarchies may be topologically
embedded, type by type, into the corresponding hierarchy over the Urysohn
space. As a preparation for this, we prove an effective density theorem that is
also of independent interest.Comment: 21 page
On the topological aspects of the theory of represented spaces
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 Escard\'o, Schr\"oder, 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.Comment: Earlier versions were titled "Compactness and separation for
represented spaces" and "A new introduction to the theory of represented
spaces
First Order Theories of Some Lattices of Open Sets
We show that the first order theory of the lattice of open sets in some
natural topological spaces is -equivalent to second order arithmetic. We
also show that for many natural computable metric spaces and computable domains
the first order theory of the lattice of effectively open sets is undecidable.
Moreover, for several important spaces (e.g., , , and the
domain ) this theory is -equivalent to first order arithmetic
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