We first prove that given any analytic filter ℱ on ω the set of all functions f on $2^{\omega }$ which can be represented as the pointwise limit relative to ℱ of some sequence ${\left(fₙ\right)}_{n\in \omega }$ of continuous functions ($f=li{m}_{ℱ}fₙ$), 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.

We continue the efforts to characterize winning strategies in various infinite games involving filters on the natural numbers in terms of combinatorial or structural properties of the given filter. Previous results in the literature included those games where player II responded with natural numbers, or finite subsets of natural numbers. In this paper we concentrate on games where player II responds with members of the dual ideal. We also give a summary of known results on filter games.

We consider two situations which relate properties of filters with properties of the limit operators with respect to these filters. In the first one, we show that the space of sequences having limits with respect to a ${\Pi }_3^0$ filter is itself ${\Pi }_3^0$ and therefore, by a result of Dobrowolski and Marciszewski, such spaces are topologically indistinguishable. This answers a question of Dobrowolski and Marciszewski. In the second one, we characterize universally measurable filters which fulfill Fatou’s lemma.

Drawing on an analogy with temporal fixpoint logic, we relate the arithmetic fixpoint definable sets to the winning positions of certain games, namely games whose winning conditions lie in the difference hierarchy over ${\Sigma }_2^0$. This both provides a simple characterization of the fixpoint hierarchy, and refines existing results on the power of the game quantifier in descriptive set theory. We raise the problem of transfinite fixpoint hierarchies.

Drawing on an analogy with temporal fixpoint logic, we relate the arithmetic fixpoint definable sets to the winning positions of certain games, namely games whose winning conditions lie in the difference hierarchy over ${\Sigma }_2^0$. This both provides a simple characterization of the fixpoint hierarchy, and refines existing results on the power of the game quantifier in descriptive set theory. We raise the problem of transfinite fixpoint hierarchies.

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...

Consider the poset $P_I=\text{Borel}\left(ℝ\right)\setminus I$ where $I$ is an arbitrary $\sigma$-ideal $\sigma$-generated by a projective collection of closed sets. Then the $P_I$ extension is given by a single real $r$ of an almost minimal degree: every real $s\in V\left[r\right]$ is Cohen-generic over $V$ or $V\left[s\right]=V\left[r\right]$.

Let X and Y be two Polish spaces. Functions f,g: X → Y are called equivalent if there exists a bijection φ from X onto itself such that g∘φ = f. Using a theorem of J. Saint Raymond we characterize functions equivalent to Borel measurable ones. This characterization answers a question asked by M. Morayne and C. Ryll-Nardzewski.

Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies $\beta \left(f\right)\le {\omega }^{\xi ₁}·{\omega }^{\xi ₂}$ for some countable ordinals ξ₁ and ξ₂ if and only if there exists a sequence (fₙ) of Baire-1 functions...