Some relatives of the Juhász Club Principle are introduced and studied in the presence of CH. In particular, it is shown that a slight strengthening of this principle implies the existence of a Suslin tree in the presence of CH.

Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{\aleph ₀}$. Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are $G_{\delta }$ has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property,...

In set theory without the Axiom of Choice ZF, we prove that for every commutative field $𝕂$, the following statement ${𝐃}_{𝕂}$: “On every non null $𝕂$-vector space, there exists a non null linear form” implies the existence of a “$𝕂$-linear extender” on every vector subspace of a $𝕂$-vector space. This solves a question raised in Morillon M., Linear forms and axioms of choice, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. In the second part of the paper, we generalize our results in the case of spherically...

We work in set-theory without choice ZF. Given a commutative field $𝕂$, we consider the statement $𝐃\left(𝕂\right)$: “On every non null $𝕂$-vector space there exists a non-null linear form.” We investigate various statements which are equivalent to $𝐃\left(𝕂\right)$ in ZF. Denoting by ${\mathbb{Z}}_2$ the two-element field, we deduce that $𝐃\left({\mathbb{Z}}_2\right)$ implies the axiom of choice for pairs. We also deduce that $𝐃\left(\mathbb{Q}\right)$ implies the axiom of choice for linearly ordered sets isomorphic with $\mathbb{Z}$.

I prove that the statement that “every linear order of size $2^{\omega }$ can be embedded in $({\omega }^{\omega },\ll )$” is consistent with MA + ¬ wKH.

We show that there is a universal control on the Szlenk index of a Lipschitz-quotient of a Banach space with countable Szlenk index. It is in particular the case when two Banach spaces are Lipschitz-homeomorphic. This provides information on the Cantor index of scattered compact sets $K$ and $L$ such that $C\left(L\right)$ is a Lipschitz-quotient of $C\left(K\right)$ (that is the case in particular when these two spaces are Lipschitz-homeomorphic). The proof requires tools of descriptive set theory.

We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional “L-like” combinatorial principles. In particular, this model satisfies the following properties: (1) ${♢}_{\delta }$ holds for every successor and Mahlo cardinal δ. (2) There is a stationary subset S of the least supercompact cardinal κ₀ such that for every δ ∈ S, ${◻}_{\delta }$ holds and δ carries a gap 1 morass. (3) A weak version of ${◻}_{\delta }$ holds for every infinite cardinal...

We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space...

Let X be a compact Hausdorff space and M a metric space. $E_0(X,M)$ is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which $E_0(X,M)$ is all of C(X,M). These include βℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of $E_0(X,M)$ as a normed linear space. We also build three first countable Eberlein...

Given an uncountable cardinal κ with $\kappa ={\kappa }^{<\kappa }$ and $2^{\kappa }$ regular, we show that there is a forcing that preserves cofinalities less than or equal to $2^{\kappa }$ and forces the existence of a well-order of H(κ⁺) that is definable over ⟨H(κ⁺),∈⟩ by a Σ₁-formula with parameters. This shows that, in contrast to the case "κ = ω", the existence of a locally definable well-order of H(κ⁺) of low complexity is consistent with failures of the GCH at κ. We also show that the forcing mentioned above introduces a Bernstein subset...

Assuming the continuum hypothesis there is an inseparable sequence of length ω₁ that contains no Lusin subsequence, while if Martin's Axiom and ¬ CH are assumed then every inseparable sequence (of length ω₁) is a union of countably many Lusin subsequences.

Under $M{A}_{{\omega }_1}$ every uncountable almost disjoint family is either anti-Luzin or has an uncountable Luzin subfamily. This fails under CH. Related properties are also investigated.