1,301 research outputs found
Computably regular topological spaces
This article continues the study of computable elementary topology started by
the author and T. Grubba in 2009 and extends the author's 2010 study of axioms
of computable separation. Several computable T3- and Tychonoff separation
axioms are introduced and their logical relation is investigated. A number of
implications between these axioms are proved and several implications are
excluded by counter examples, however, many questions have not yet been
answered. Known results on computable metrization of T3-spaces from M.
Schr/"oder (1998) and T. Grubba, M. Schr/"oder and the author (2007) are proved
under uniform assumptions and with partly simpler proofs, in particular, the
theorem that every computably regular computable topological space with
non-empty base elements can be embedded into a computable metric space. Most of
the computable separation axioms remain true for finite products of spaces
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
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
Computable Separation in Topology, from T_0 to T_3
This article continues the study of computable elementary topology started in (Weihrauch, Grubba 2009). We introduce a number of computable versions of the topological to separation axioms and solve their logical relation completely. In particular, it turns out that computable is equivalent to computable . The strongest axiom is used in (Grubba, Schroeder, Weihrauch 2007) to construct a computable metric
Quasi-Polish Spaces
We investigate some basic descriptive set theory for countably based
completely quasi-metrizable topological spaces, which we refer to as
quasi-Polish spaces. These spaces naturally generalize much of the classical
descriptive set theory of Polish spaces to the non-Hausdorff setting. We show
that a subspace of a quasi-Polish space is quasi-Polish if and only if it is
level \Pi_2 in the Borel hierarchy. Quasi-Polish spaces can be characterized
within the framework of Type-2 Theory of Effectivity as precisely the countably
based spaces that have an admissible representation with a Polish domain. They
can also be characterized domain theoretically as precisely the spaces that are
homeomorphic to the subspace of all non-compact elements of an
\omega-continuous domain. Every countably based locally compact sober space is
quasi-Polish, hence every \omega-continuous domain is quasi-Polish. A
metrizable space is quasi-Polish if and only if it is Polish. We show that the
Borel hierarchy on an uncountable quasi-Polish space does not collapse, and
that the Hausdorff-Kuratowski theorem generalizes to all quasi-Polish spaces
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