Order topology and bi-Scott topology on a poset

In this paper, some properties of order topology and bi-Scott topology on a poset are obtained. Order-convergence in posets is further studied. Especially, a sufficient and necessary condition for order-convergence to be topological is given for some kind of posets.     

L-fuzzifying topology

A new topology in terms of order on fuzzy sets, revealing better the relationship between smooth topology and Chang’s fuzzy topology, is presented in the paper. Some basic properties are discussed.     

Tropical topology

A space-periodic ground state is shown to exist for lattices of point ions in R³ coupled to the Schrödinger and scalar fields. The coupling requires renormalization due to the singularity of the Coulomb self-action. The ground state is constructed by minimizing the renormalized energy per cell. This energy is bounded from below when the charge of each ion is positive. The elementary cell is necessarily neutral.     

The finest Nachbin topology is a mixed topology

Let T be a completely regular topological space. We show that the finest Nachbin topology on C(T) is the finest locally convex topology on C(T) that coincides with the compact open topology on the order bounded intervals of C(T).     

Soft topology

In this paper, the notions of topology on soft subsets and soft topology have been introduced. Some basic properties of these topologies are studied. Also the definition of continuity of soft mappings in both the topologies is given with their properties studied.     

Ψ-density topology

We study the notion of ψ-density point and ψ-density topologies for non-decreasing continuous functions ψ. Also basic properties of ψ-approximately continuous functions are considered.     

Simple density topology

We introduce a new topology on the real line generated by the simple density points for measure. We show also that a simple category density point does not lead to a new notion. Supported by research project “Analisi Reale”, Italian PRIN funds Visiting Professor at Facoltà di Economia, Università Federico II, Naples, Italy     

Towards higher topology

We categorify the adjunction between locales and topological spaces, this amounts to an adjunction between (generalized) bounded ionads and topoi. We show that the adjunction is idempotent. We relate this adjunction to the Scott adjunction, which was discussed from a more categorical point of view in [21]. We hint that 0-dimensional adjunction inhabits the categorified one.     

Monads in topology

Let T be a submonad of the ultrafilter monad β and let G be a subfunctor of the filter functor. The T-algebras are topological spaces whose closed sets are the subalgebras and form thereby an equationally definable full subcategory of topological spaces. For appropriate T, countably generated free algebras provide ZFC examples of separable, Urysohn, countably compact, countably tight spaces which are neither compact nor sequential, and c² non-homeomorphic such examples exist. For any space X, say that U⊂X is G-open if U belongs to every ultrafilter in GX which converges in U. The full subcategory TopG consists of all G-spaces, those spaces in which every G-open set is open. Each TopG has at least these stability properties: it contains all Alexandroff spaces, and is closed under coproducts, quotients and locally closed subspaces. Examples include sequential spaces, P-spaces and countably tight spaces. T-algebras are characterized as the T-compact, T-Hausdorff T-spaces. Malyhin's theorem on countable tightness generalizes verbatim to TopG for any G⊂β. For r∈ω^?=βω\ω, let Gr be the subfunctor of β generated by r and let Tr be the generated submonad. If RK_? is the Rudin-Keisler preorder on ω^?, rRK_?s⇔Gr⊂Gs. Let c_? be the Comfort preorder and define the monadic preorderrm_?s to mean Tr⊂Ts. Then rRK_?s⇒rm_?s⇒rc_?s. It follows that there exist c² monadic types. For each such type Tr, the Tr-algebras form an equationally definable full subcategory of topological spaces with only one operation of countably infinite arity. No two of these varieties are term equivalent nor is any one a full subcategory of another inside topological spaces. Say that r∈ω^? is an m-point if Gr≠Tr. Under CH, m-points exist.     

Constructive algebraic topology

The classical “computation” methods in Algebraic Topology most often work by means of highly infinite objects and in fact are not constructive. Typical examples are shown to describe the nature of the problem. The Rubio-Sergeraert solution for Constructive Algebraic Topology is recalled. This is not only a theoretical solution: the concrete computer program Kenzo has been written down which precisely follows this method. This program has been used in various cases, opening new research subjects and producing in several cases significant results unreachable by hand. In particular the Kenzo program can compute the first homotopy groups of a simply connected arbitrary simplicial set.     

Distortion and topology

For a self mapping f: D→D of the unit disk in C which has finite distortion, we give a separation condition on the components of the set where the distortion is very large - say greater than a given constant - which implies that f still extends homeomorphically and quasisymmetrically to the boundary S = ?D. Thus f shares its boundary values with a quasiconformal mapping whose distortion we explicitly estimate in terms of the data. This condition, uniformly separated in modulus, allows the set where the distortion is large to accumulate on the entire boundary S, but it does not allow a component to run out to the boundary - a necessary restriction. The lift of a Jordan domain in a Riemann surface to its universal cover D is always uniformly separated in modulus, and this allows us to apply these results in the theory of Riemann surfaces to identify an interesting link between the support of the high distortion of a map between surfaces and their geometry - again with explicit estimates. As part of our investigations, we study mappings ?: S → S which are the germs of a conformal mapping and give good bounds on the distortion of a quasiconformal extension of ? to the disk D. We then extend these results to the germs of quasisymmetric mappings. These appear of independent interest and identify new geometric invariants.     

Complete density topology

Using the notion of the complete convergence of a sequence of measurable functions we introduce the notion of a complete density point of a measurable set. Using complete density points we generate a topology on the real line between ordinary and density topology. An ingenious construction of Lekkerkerker enables us to prove that the simple density topology is strictly stronger than the complete topology.