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Filter descriptive classes of Borel functions

Gabriel Debs, Jean Saint Raymond (2009)

Fundamenta Mathematicae

We first prove that given any analytic filter ℱ on ω the set of all functions f on 2 ω which can be represented as the pointwise limit relative to ℱ of some sequence ( f ₙ ) n ∈ ω of continuous functions ( f = l i m ℱ f ₙ ), is exactly the set of all Borel functions of class ξ for some countable ordinal ξ that we call the rank of ℱ. We discuss several structural properties of this rank. For example, we prove that any free Π⁰₄ filter is of rank 1.

Finite-to-one fuzzy maps and fuzzy perfect maps

Francisco Gallego Lupiañez (1998)

Kybernetika

In this paper we define, for fuzzy topology, notions corresponding to finite-to-one and k -to-one maps. We study the relationship between these new fuzzy maps and various kinds of fuzzy perfect maps. Also, we show the invariance and the inverse inveriance under the various kinds of fuzzy perfect maps (and the finite-to-one fuzzy maps), of different properties of fuzzy topological spaces.

First countable spaces without point-countable π-bases

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy (2007)

Fundamenta Mathematicae

We answer several questions of V. Tkachuk [Fund. Math. 186 (2005)] by showing that ∙ there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable π-base (in fact, the minimum order of a π-base of the space can be made arbitrarily large); ∙ if there is a κ-Suslin line then there is a first countable GO-space of cardinality κ⁺ in which the order of any π-base is at least κ; ∙ it is consistent to have a first countable,...

Fixed point theorems of G -fuzzy contractions in fuzzy metric spaces endowed with a graph

Satish Shukla (2014)

Communications in Mathematics

Let ( X , M , * ) be a fuzzy metric space endowed with a graph G such that the set V ( G ) of vertices of G coincides with X . Then we define a G -fuzzy contraction on X and prove some results concerning the existence and uniqueness of fixed point for such mappings. As a consequence of the main results we derive some extensions of known results from metric into fuzzy metric spaces. Some examples are given which illustrate the results.

F-limit points in dynamical systems defined on the interval

Piotr Szuca (2013)

Open Mathematics

Given a free ultrafilter p on ℕ we say that x ∈ [0, 1] is the p-limit point of a sequence (x n)n∈ℕ ⊂ [0, 1] (in symbols, x = p -limn∈ℕ x n) if for every neighbourhood V of x, {n ∈ ℕ: x n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f p: [0, 1] → [0, 1] is defined by f p(x) = p -limn∈ℕ f n(x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f p. For a filter F we also define the ω F-limit set of f at x. We consider...

Forcing countable networks for spaces satisfying R ( X ω ) = ω

István Juhász, Lajos Soukup, Zoltán Szentmiklóssy (1996)

Commentationes Mathematicae Universitatis Carolinae

We show that all finite powers of a Hausdorff space X do not contain uncountable weakly separated subspaces iff there is a c.c.c poset P such that in V P X is a countable union of 0 -dimensional subspaces of countable weight. We also show that this...

Forcing tightness in products of fans

Jörg Brendle, Tim La Berge (1996)

Fundamenta Mathematicae

We prove two theorems that characterize tightness in certain products of fans in terms of families of integer-valued functions. We also define several notions of forcing that allow us to manipulate the structure of the set of functions from some cardinal θ to ω, and hence, the tightness of these products. These results give new constructions of first countable <θ-cwH spaces that are not ≤θ-cwH.

Function space topologies deriving from hypertopologies and networks

A. Di Concilio, A. Miranda (2001)

Bollettino dell'Unione Matematica Italiana

In un progetto di generalizzazione delle classiche topologie di tipo «set-open» di Arens-Dugundji introduciamo un metodo generale per produrre topologie in spazi di funzioni mediante l'uso di ipertopologie. Siano X , Y spazi topologici e C ⁢ X , Y l'insieme delle funzioni continue da X verso Y . Fissato un «network» α nel dominio X ed una topologia τ nell'iperspazio C ⁢ L ⁢ Y del codominio Y si genera una topologia τ α in C ⁢ X , Y richiedendo che una rete f λ di C ⁢ X , Y converge in τ α ad f ∈ C ⁢ X , Y se e solo se la rete f λ ⁢ A ¯ converge in τ ad f ⁢ A ¯ ...

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