Related papers: A bump in the road in elementary topology
The paper approaches to the reader to topology with a curious example of topological classification by homeomorphisms applied to the letters of the alphabet viewed as subsets of Euclidean plane.
Rough set theory is a new mathematical approach to imperfect knowledge. The notion of rough sets is generalized by using an arbitrary binary relation on attribute values in information systems, instead of the trivial equality relation. The…
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…
We introduce a notion of a weak elementary fibration and prove that it does exist in certain interesting cases. Our notion is a modification of the M. Artin's notion of an elementary fibration.
A topology is defined on the mapping class group of a compact connected orientable surface. It is shown that a notion of "genericity" on subsets of the mapping class group arises from this definition. Many plausible results follow from this…
This article is dedicated to the investigation of difficulties involved in the understanding of the homomorphism concept. It doesn't restrict to group-theory but on the contrary raises the issue of developing teaching strategies aiming at…
A reinterpretation of noncommutativity as a mapping of paths is proposed at the level of quantum mechanics.
We consider the problem of distance between two particles in the universe, where space is taken to be Liebnizian rather than Newtonian, this being the present day approach. We then argue that with latest inputs from physics, it is possible…
It it shown that geometric morphisms between elementary toposes can be represented as adjunctions between the corresponding categories of locales. These adjunctions are characterised as those that preserve the order enrichment, commute with…
We introduce the notion of negative topological dimension and the notion of weight for the asymptotic topological dimension. Quantizing of spaces of negative dimension is applied to linguistic statistics.
The purpose of this article is to investigate the relationship between suborbifolds and orbifold embeddings. In particular, we give natural definitions of the notion of suborbifold and orbifold embedding and provide many examples.…
In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to…
We rewrite classical topological definitions using the category-theoretic notation of arrows and are led to concise reformulations in terms of simplicial categories and orthogonality of morphisms, which we hope might be of use in the…
In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semi-classical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex…
In any attempt to build a quantum theory of gravity, a central issue is to unravel the structure of space-time at the smallest scale. Of particular relevance is the possible definition of coordinate functions within the theory and the study…
We construct a function for almost-complex Riemannian manifolds. Non-vanishing of the function for the almost-complex structure implies the almost-complex structure is not integrable. Therefore the constructed function is an obstruction for…
We describe various strengthenings of the concept of topological transitivity. Especially when one departs from the family of invertible systems, a number of interesting properties arise. We present the architecture of implications among…
The Morse-Bott inequalities relate the topology of a closed manifold to the topology of the critical point set of a Morse-Bott function defined on it. The Morse-Bott inequalities are sometimes stated under incorrect orientation assumptions.…
Given a functor $p:E \rightarrow B$ and an object $e \in E$ , we define a \emph{displacement} of $e$ along a morphism $\varepsilon: p(e) \rightarrow b$, as a map $e \rightarrow \nabla_\varepsilon(e)$ satisfying a universal property analogue…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part we discus the main structures…