41 research outputs found
The reals as rational Cauchy filters
We present a detailed and elementary construction of the real numbers from
the rational numbers a la Bourbaki. The real numbers are defined to be the set
of all minimal Cauchy filters in (where the Cauchy condition is
defined in terms of the absolute value function on ) and are proven
directly, without employing any of the techniques of uniform spaces, to form a
complete ordered field. The construction can be seen as a variant of Bachmann's
construction by means of nested rational intervals, allowing for a canonical
choice of representatives
Metric characterization of connectedness for topological spaces
Connectedness, path connectedness, and uniform connectedness are well-known
concepts. In the traditional presentation of these concepts there is a
substantial difference between connectedness and the other two notions, namely
connectedness is defined as the absence of disconnectedness, while path
connectedness and uniform connectedness are defined in terms of connecting
paths and connecting chains, respectively. In compact metric spaces uniform
connectedness and connectedness are well-known to coincide, thus the apparent
conceptual difference between the two notions disappears. Connectedness in
topological spaces can also be defined in terms of chains governed by open
coverings in a manner that is more reminiscent of path connectedness. We
present a unifying metric formalism for connectedness, which encompasses both
connectedness of topological spaces and uniform connectedness of uniform
spaces, and which further extends to a hierarchy of notions of connectedness
A folk Quillen model structure for operads
We establish, by elementary means, the existence of a cofibrantly generated
monoidal model structure on the category of operads. By slicing over a suitable
operad the classical Rezk model structure on the category of small categories
is recovered
The real numbers - a survey of constructions
We present a comprehensive survey of constructions of the real numbers (from
either the rationals or the integers) in a unified fashion, thus providing an
overview of most (if not all) known constructions ranging from the earliest
attempts to recent results, and allowing for a simple comparison-at-a-glance
between different constructions
Metric 1-spaces
A generalization of metric space is presented which is shown to admit a
theory strongly related to that of ordinary metric spaces. To avoid the
topological effects related to dropping any of the axioms of metric space,
first a new, and equivalent, axiomatization of metric space is given which is
then generalized from a fresh point of view. Naturally arising examples from
metric geometry are presented
An ordered framework for partial multivalued functors
The category Rel of sets and relations intimately ties the notions of
function, partial multivalued function, and direct image under a function
through the description of Rel as the Kleisli category of the covariant power
set functor on Set. We present a suitable framework to obtain a similar
relationship between the concepts of functor, partial multivalued functor, and
the direct image under a functor.Comment: Accepted for presentation at the Asia-Pacific World Congress on
Computer Science and Engineering 2015, Fij
Completion of continuity spaces with uniformly vanishing asymmetry
The classical Cauchy completion of a metric space (by means of Cauchy
sequences) as well as the completion of a uniform space (by means of Cauchy
filters) are well-known to rely on the symmetry of the metric space or uniform
space in question. For qausi-metric spaces and quasi-uniform spaces various
non-equivalent completions exist, often defined on a certain subcategory of
spaces that satisfy a key property required for the particular completion to
exist. The classical filter completion of a uniform space can be adapted to
yield a filter completion of a metric space. We show that this completion by
filters generalizes to continuity spaces that satisfy a form of symmetry which
we call uniformly vanishing asymmetry