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On the Intuitionistic fuzzy topological (metric and normed) spaces
In this paper, we define precompact set in intuitionistic fuzzy metric spaces
and prove that any subset of an intuitionistic fuzzy metric space is compact if
and only if it is precompact and complete. Also we define topologically
complete intuitionistic fuzzy metrizable spaces and prove that any set in a complete intuitionistic fuzzy metric spaces is a topologically
complete intuitionistic fuzzy metrizable space and vice versa. Finally, we
define intuitionistic fuzzy normed spaces and fuzzy boundedness for linear
operators and so we prove that every finite dimensional intuitionistic fuzzy
normed space is complete.Comment: 16 page
Spectral geometry with a cut-off: topological and metric aspects
Inspired by regularization in quantum field theory, we study topological and
metric properties of spaces in which a cut-off is introduced. We work in the
framework of noncommutative geometry, and focus on Connes distance associated
to a spectral triple (A, H, D). A high momentum (short distance) cut-off is
implemented by the action of a projection P on the Dirac operator D and/or on
the algebra A. This action induces two new distances. We individuate conditions
making them equivalent to the original distance. We also study the
Gromov-Hausdorff limit of the set of truncated states, first for compact
quantum metric spaces in the sense of Rieffel, then for arbitrary spectral
triples. To this aim, we introduce a notion of "state with finite moment of
order 1" for noncommutative algebras. We then focus on the commutative case,
and show that the cut-off induces a minimal length between points, which is
infinite if P has finite rank. When P is a spectral projection of , we work
out an approximation of points by non-pure states that are at finite distance
from each other. On the circle, such approximations are given by Fejer
probability distributions. Finally we apply the results to Moyal plane and the
fuzzy sphere, obtained as Berezin quantization of the plane and the sphere
respectively.Comment: Reference added. Minor corrections. Published version. 38 pages, 2
figures. Journal of Geometry and Physics 201
Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory
In Riesz space theory it is good practice to avoid representation theorems
which depend on the axiom of choice. Here we present a general methodology to
do this using pointfree topology. To illustrate the technique we show that
almost f-algebras are commutative. The proof is obtained relatively
straightforward from the proof by Buskes and van Rooij by using the pointfree
Stone-Yosida representation theorem by Coquand and Spitters
Spectral C*-categories and Fell bundles with path-lifting
Following Crane's suggestion that categorification should be of fundamental
importance in quantising gravity, we show that finite dimensional even
-real spectral triples over \bbc are already nothing more than full
C*-categories together with a self-adjoint section of their domain and range
maps, while the latter are equivalent to unital saturated Fell bundles over
pair groupoids equipped with a path-lifting operator given by a normaliser.
Interpretations can be made in the direction of quantum Higgs gravity. These
geometries are automatically quantum geometries and we reconstruct the
classical limit, that is, general relativity on a Riemannian spin manifold.Comment: 20 pages, 1 figur
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