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Double Sequences and Iterated Limits in Regular Space

Roland Coghetto (2016)

Formalized Mathematics

First, we define in Mizar [5], the Cartesian product of two filters bases and the Cartesian product of two filters. After comparing the product of two Fréchet filters on ℕ (F1) with the Fréchet filter on ℕ × ℕ (F2), we compare limF₁ and limF₂ for all double sequences in a non empty topological space. Endou, Okazaki and Shidama formalized in [14] the “convergence in Pringsheim’s sense” for double sequence of real numbers. We show some basic correspondences between the p-convergence and the filter...

Examples of sequential topological groups under the continuum hypothesis

Alexander Shibakov (1996)

Fundamenta Mathematicae

Using CH we construct examples of sequential topological groups: 1. a pair of countable Fréchet topological groups whose product is sequential but is not Fréchet, 2. a countable Fréchet and α 1 topological group which contains no copy of the rationals.

Extension of multisequences and countably uniradial classes of topologies

Szymon Dolecki, Andrzej Starosolski, Stephen W. Watson (2003)

Commentationes Mathematicae Universitatis Carolinae

It is proved that every non trivial continuous map between the sets of extremal elements of monotone sequential cascades can be continuously extended to some subcascades. This implies a result of Franklin and Rajagopalan that an Arens space cannot be continuously non trivially mapped to an Arens space of higher rank. As an application, it is proved that if for a filter ℋ on ω , the class of ℋ -radial topologies contains each sequential topology, then it includes the class of subsequential topologies....

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.

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...

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