We prove that the class of functions with the Baire property has the weak difference property in category sense. That is, every function for which f(x+h)−f(x) has the Baire property for every h ∈ R can be written in the form f = g+H+φ where g has the Baire property, H is additive, and for every h ∈ R we have φ(x+h)−φ(x) 6= 0 only on a meager set. We also discuss the weak difference property of ...

The main result is the following. Let f:X→Y be a continuous mapping of completely Baire space X onto hereditary weakly Preiss-Simon regular Y such that image every open subset resolvable set in Y. Then Baire. classical Hurewicz theorem about closed embedding rational numbers into metrizable spaces generalized to spaces.

In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rati...

We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (Σ 2 and, under suitable restrictions, Π 2 ) are shown to have pleasant properties, related to Baire category. We construct models of set theory where (unrestricted) Π 2 -characteristics behave quite chaotically and no...

Proof If Cexp is model complete, then every definable set is a projection of a closed set. Since C is locally compact, every definable set is Fσ. The same is true for the complement, so every definable set is also Gδ. But, since Z is definable, Q is definable and a standard corollary of the Baire Category Theorem tells us that Q is not Gδ. Still, there are several interesting open questions abo...

The question of whether a hypercyclic operator T acting on Fréchet algebra X admits or not an vectors (but 0) has been addressed in the recent literature. In this paper we give new criteria and characterisations context convolution operators H(C) backward shifts general sequence algebra. Analogous questions arise for stronger properties like frequent hypercyclicity. trend sufficient condition w...

In this paper, we study soft sets of the first and second Baire categories. The category are examined to be small from point view topology, while large. family in a topological space forms σ-ideal. This contributes development theory ideal topology. main properties these classes discussed. concepts points where or introduced. These types subclasses non-cluster cluster sets. Then, various result...

In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements are X , ω1,...