Abstract If is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable , no member of which is covered by finitely many functions from , there is such that for all there are infinitely many integersksuch thatf(k) = h(k). However ifV = Lthen there exists a coanalytic family of pairwise eventually different functions satisfying this strong maximality condition.

Abstract In this paper we show that it is consistent with ZFC that the cardinality of every maximal cofinitary group of Sym(ω) is strictly greater than the cardinal numbers and .

Abstract The Mycielski ideal is defined to consist of all setsA⊆ℕksuch that {f↾X:f∈A} ≠Xkfor all . It will be shown that the covering numbers for these ideals are all equal. However, the covering numbers of the closely associated Rosłanowski ideals will be shown to be consistently different.

Abstract To any metric space it is possible to associate the cardinal invariant corresponding to the least number of rectifiable curves in the space whose union is not meagre. It is shown that this invariant can vary with the metric space considered, even when restricted to the class of convex subspaces of separable Banach spaces. As a corollary it is obtained that it is consistent with set theory that any set of reals of size ℵ1 is meagre yet there are ℵ1 rectifiable curves in ℝ3 whose union is not meagre. The consistency of this statement when the phrase “rectifiable curves” is replaced by “straight lines” remains open.

For x, y ϵ ℝω define the inner product which may not be finite or even exist. We say that x and y are orthogonal if (x, y) converges and equals 0. Define lp to be the set of all x ϵ ℝω such that For Hilbert space, l2, any family of pairwise orthogonal sequences must be countable. For a good introduction to Hilbert space, see Retherford [4]. Theorem 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of lp for every p > 2. It was already known that there exists a family of continuum many pairwise orthogonal elements of ℝω. A family F ⊆ ℝω∖0 of pairwise orthogonal sequences is orthogonally complete or a maximal orthogonal family iff the only element of ℝω orthogonal to every element of F is 0, the constant 0 sequence. It is somewhat surprising that Kunen's perfect set of orthogonal elements is maximal (a fact first asserted by Abian). MAD families, nonprincipal ultrafilters, and many other such maximal objects cannot be even Borel. Theorem 2. There exists a perfect maximal orthogonal family of elements of ℝω. Abian raised the question of what are the possible cardinalities of maximal orthogonal families. Theorem 3. In the Cohen real model there is a maximal orthogonal set in ℝω of cardinality ω1, but there is no maximal orthogonal set of cardinality κ with ω1 < κ < ϲ. By the Cohen real model we mean any model obtained by forcing with finite partial functions from γ to 2, where the ground model satisfies GCH and γω = γ.

Abstract By ℬ2 we denote the σ-ideal of all subsets A of the Cantor set {0, 1}ω such that for every infinite subset T of ω the restriction A∣{0, 1}T is a proper subset of {0, 1}T. In this paper we investigate set theoretical properties of this and similar ideals.

Abstract It is shown to be consistent that the reals are covered by ℵ1, meagre sets yet there is a Baire class 1 function which cannot be covered by fewer than ℵ2, continuous functions. A new cardinal invariant is introduced which corresponds to the least number of continuous functions required to cover a given function. This is characterized combinatorially. A forcing notion similar to, but not equivalent to, superperfect forcing is introduced.