123,497 research outputs found
The principle of pointfree continuity
In the setting of constructive pointfree topology, we introduce a notion of
continuous operation between pointfree topologies and the corresponding
principle of pointfree continuity. An operation between points of pointfree
topologies is continuous if it is induced by a relation between the bases of
the topologies; this gives a rigorous condition for Brouwer's continuity
principle to hold. The principle of pointfree continuity for pointfree
topologies and says that any relation which induces
a continuous operation between points is a morphism from to
. The principle holds under the assumption of bi-spatiality of
. When is the formal Baire space or the formal unit
interval and is the formal topology of natural numbers, the
principle is equivalent to spatiality of the formal Baire space and formal unit
interval, respectively. Some of the well-known connections between spatiality,
bar induction, and compactness of the unit interval are recast in terms of our
principle of continuity.
We adopt the Minimalist Foundation as our constructive foundation, and
positive topology as the notion of pointfree topology. This allows us to
distinguish ideal objects from constructive ones, and in particular, to
interpret choice sequences as points of the formal Baire space
Positivity relations on a locale
This paper analyses the notion of a positivity relationof Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finally, some connection is revealed between positivity relations and localic suplattices (these are algebras for the powerlocale monad)
Model Checking Spatial Logics for Closure Spaces
Spatial aspects of computation are becoming increasingly relevant in Computer
Science, especially in the field of collective adaptive systems and when
dealing with systems distributed in physical space. Traditional formal
verification techniques are well suited to analyse the temporal evolution of
programs; however, properties of space are typically not taken into account
explicitly. We present a topology-based approach to formal verification of
spatial properties depending upon physical space. We define an appropriate
logic, stemming from the tradition of topological interpretations of modal
logics, dating back to earlier logicians such as Tarski, where modalities
describe neighbourhood. We lift the topological definitions to the more general
setting of closure spaces, also encompassing discrete, graph-based structures.
We extend the framework with a spatial surrounded operator, a propagation
operator and with some collective operators. The latter are interpreted over
arbitrary sets of points instead of individual points in space. We define
efficient model checking procedures, both for the individual and the collective
spatial fragments of the logic and provide a proof-of-concept tool
Localic completion of uniform spaces
We extend the notion of localic completion of generalised metric spaces by
Steven Vickers to the setting of generalised uniform spaces. A generalised
uniform space (gus) is a set X equipped with a family of generalised metrics on
X, where a generalised metric on X is a map from the product of X to the upper
reals satisfying zero self-distance law and triangle inequality.
For a symmetric generalised uniform space, the localic completion lifts its
generalised uniform structure to a point-free generalised uniform structure.
This point-free structure induces a complete generalised uniform structure on
the set of formal points of the localic completion that gives the standard
completion of the original gus with Cauchy filters.
We extend the localic completion to a full and faithful functor from the
category of locally compact uniform spaces into that of overt locally compact
completely regular formal topologies. Moreover, we give an elementary
characterisation of the cover of the localic completion of a locally compact
uniform space that simplifies the existing characterisation for metric spaces.
These results generalise the corresponding results for metric spaces by Erik
Palmgren.
Furthermore, we show that the localic completion of a symmetric gus is
equivalent to the point-free completion of the uniform formal topology
associated with the gus.
We work in Aczel's constructive set theory CZF with the Regular Extension
Axiom. Some of our results also require Countable Choice.Comment: 39 page
Hyperspaces in topological Categories
Hyperspaces form a powerful tool in some branches of mathematics: lots of
fractal and other geometric objects can be viewed as fixed points of some
functions in suitable hyperspaces - as well as interesting classes of formal
languages in theoretical computer sciences, for example (to illustrate the wide
scope of this concept). Moreover, there are many connections between
hyperspaces and function spaces in topology. Thus results from hyperspaces help
to get new results in function spaces and vice versa. Unfortunately, there ist
no natural hyperspace construction known for general topological categories (in
contrast to the situation for function spaces). We will shortly present a
rather combinatorial idea for the transfer of structure from a set to a
subset of , just to motivate an interesting question in set
theory. Then we will propose and discuss a new approach to define
hyperstructures, which works in every cartesian closed topological category,
and so applies to every topological category, using it's topological universe
hull
A Few Notes on Formal Balls
Using the notion of formal ball, we present a few new results in the theory
of quasi-metric spaces. With no specific order: every continuous
Yoneda-complete quasi-metric space is sober and convergence Choquet-complete
hence Baire in its -Scott topology; for standard quasi-metric spaces,
algebraicity is equivalent to having enough center points; on a standard
quasi-metric space, every lower semicontinuous -valued
function is the supremum of a chain of Lipschitz Yoneda-continuous maps; the
continuous Yoneda-complete quasi-metric spaces are exactly the retracts of
algebraic Yoneda-complete quasi-metric spaces; every continuous Yoneda-complete
quasi-metric space has a so-called quasi-ideal model, generalizing a
construction due to K. Martin. The point is that all those results reduce to
domain-theoretic constructions on posets of formal balls
Density of classical points in eigenvarieties
In this short note, we study the geometry of the eigenvariety parametrizing p-adic automorphic forms for GL(1) over a number field, as constructed by Buzzard. We show that if K is not totally real and contains no CM subfield, points in this space arising from classical automorphic forms (i.e. algebraic Grossencharacters of K) are not Zariski-dense in the eigenvariety (as a rigid space); but the eigenvariety posesses a natural formal scheme model, and the set of classical points is Zariski-dense in the formal scheme.
We also sketch the theory for GL(2) over an imaginary quadratic field, following Calegari and Mazur, emphasizing the strong formal similarity with the case of GL(1) over a general number field
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