1,917 research outputs found
Continuous reducibility and dimension of metric spaces
If is a Polish metric space of dimension , then by Wadge's lemma,
no more than two Borel subsets of can be incomparable with respect to
continuous reducibility. In contrast, our main result shows that for any metric
space of positive dimension, there are uncountably many Borel subsets
of that are pairwise incomparable with respect to continuous
reducibility.
The reducibility that is given by the collection of continuous functions on a
topological space is called the \emph{Wadge quasi-order} for
. We further show that this quasi-order, restricted to the Borel
subsets of a Polish space , is a \emph{well-quasiorder (wqo)} if and
only if has dimension , as an application of the main result.
Moreover, we give further examples of applications of the technique, which is
based on a construction of graph colorings
The descriptive theory of represented spaces
This is a survey on the ongoing development of a descriptive theory of
represented spaces, which is intended as an extension of both classical and
effective descriptive set theory to deal with both sets and functions between
represented spaces. Most material is from work-in-progress, and thus there may
be a stronger focus on projects involving the author than an objective survey
would merit.Comment: survey of work-in-progres
Connected Choice and the Brouwer Fixed Point Theorem
We study the computational content of the Brouwer Fixed Point Theorem in the
Weihrauch lattice. Connected choice is the operation that finds a point in a
non-empty connected closed set given by negative information. One of our main
results is that for any fixed dimension the Brouwer Fixed Point Theorem of that
dimension is computably equivalent to connected choice of the Euclidean unit
cube of the same dimension. Another main result is that connected choice is
complete for dimension greater than or equal to two in the sense that it is
computably equivalent to Weak K\H{o}nig's Lemma. While we can present two
independent proofs for dimension three and upwards that are either based on a
simple geometric construction or a combinatorial argument, the proof for
dimension two is based on a more involved inverse limit construction. The
connected choice operation in dimension one is known to be equivalent to the
Intermediate Value Theorem; we prove that this problem is not idempotent in
contrast to the case of dimension two and upwards. We also prove that Lipschitz
continuity with Lipschitz constants strictly larger than one does not simplify
finding fixed points. Finally, we prove that finding a connectedness component
of a closed subset of the Euclidean unit cube of any dimension greater or equal
to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these
results, we introduce a representation of closed subsets of the unit cube by
trees of rational complexes.Comment: 36 page
Lipschitz and uniformly continuous reducibilities on ultrametric Polish spaces
We analyze the reducibilities induced by, respectively, uniformly continuous,
Lipschitz, and nonexpansive functions on arbitrary ultrametric Polish spaces,
and determine whether under suitable set-theoretical assumptions the induced
degree-structures are well-behaved.Comment: 37 pages, 2 figures, revised version, accepted for publication in the
Festschrift that will be published on the occasion of Victor Selivanov's 60th
birthday by Ontos-Verlag. A mistake has been corrected in Section
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