24,929 research outputs found
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
Levels of discontinuity, limit-computability, and jump operators
We develop a general theory of jump operators, which is intended to provide
an abstraction of the notion of "limit-computability" on represented spaces.
Jump operators also provide a framework with a strong categorical flavor for
investigating degrees of discontinuity of functions and hierarchies of sets on
represented spaces. We will provide a thorough investigation within this
framework of a hierarchy of -measurable functions between arbitrary
countably based -spaces, which captures the notion of computing with
ordinal mind-change bounds. Our abstract approach not only raises new questions
but also sheds new light on previous results. For example, we introduce a
notion of "higher order" descriptive set theoretical objects, we generalize a
recent characterization of the computability theoretic notion of "lowness" in
terms of adjoint functors, and we show that our framework encompasses ordinal
quantifications of the non-constructiveness of Hilbert's finite basis theorem
A generalization of a theorem of Hurewicz for quasi-Polish spaces
We identify four countable topological spaces , , , and
which serve as canonical examples of topological spaces which fail to be
quasi-Polish. These four spaces respectively correspond to the , ,
, and -separation axioms. is the space of rationals, is
the natural numbers with the cofinite topology, is an infinite chain
without a top element, and is the set of finite sequences of natural
numbers with the lower topology induced by the prefix ordering. Our main result
is a generalization of Hurewicz's theorem showing that a co-analytic subset of
a quasi-Polish space is either quasi-Polish or else contains a countable
-subset homeomorphic to one of these four spaces
On the commutativity of the powerspace constructions
We investigate powerspace constructions on topological spaces, with a
particular focus on the category of quasi-Polish spaces. We show that the upper
and lower powerspaces commute on all quasi-Polish spaces, and show more
generally that this commutativity is equivalent to the topological property of
consonance. We then investigate powerspace constructions on the open set
lattices of quasi-Polish spaces, and provide a complete characterization of how
the upper and lower powerspaces distribute over the open set lattice
construction
Special curves and postcritically-finite polynomials
We study the postcritically-finite (PCF) maps in the moduli space of complex
polynomials . For a certain class of rational curves in
, we characterize the condition that contains infinitely
many PCF maps. In particular, we show that if is parameterized by
polynomials, then there are infinitely many PCF maps in if and only if
there is exactly one active critical point along , up to symmetries; we
provide the critical orbit relation satisfied by any pair of active critical
points. For the curves in the space of cubic
polynomials, introduced by Milnor (1992), we show that
contains infinitely many PCF maps if and only if
. The proofs involve a combination of number-theoretic methods
(specifically, arithmetic equidistribution) and complex-analytic techniques
(specifically, univalent function theory). We provide a conjecture about
Zariski density of PCF maps in subvarieties of the space of rational maps, in
analogy with the Andr\'e-Oort Conjecture from arithmetic geometry.Comment: Final version, appeared in Forum of Math. P
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