Given an ordered metric space (in particular, a Banach lattice) E, the generalized Helly space H(E) is the set of all increasing functions from the interval [0,1] to E considered with the topology of pointwise convergence, and E is said to have property (λ) if each of these functions has only countably many points of discontinuity. The main objective of the paper is to study those ordered metric spaces C(K,E), where K is a compact space, that have property (λ). In doing so, the guiding idea comes...

We construct a precompact completely regular paratopological Abelian group G of size (2ω)+ such that all subsets of G of cardinality ≤ 2ω are closed. This shows that Protasov’s theorem on non-closed discrete subsets of precompact topological groups cannot be extended to paratopological groups. We also prove that the group reflection of the product of an arbitrary family of paratopological (even semitopological) groups is topologically isomorphic to the product of the group reflections of the factors,...

We deal with a hyperspace selection problem in the setting of connected spaces. We present two solutions of this problem illustrating the difference between selections for the nonempty closed sets, and those for the at most two-point sets. In the first case, we obtain a characterisation of compact orderable spaces. In the latter case --- that of selections for at most two-point sets, the same selection property is equivalent to the existence of a ternary relation on the space, known as a cyclic...

We call a topological space $\kappa$-compact if every subset of size $\kappa$ has a complete accumulation point in it. Let $\Phi (\mu ,\kappa ,\lambda )$ denote the following statement: $\mu <\kappa <\lambda =cf\left(\lambda \right)$ and there is $\{S_{\xi }:\xi <\lambda \}\subset {\left[\kappa \right]}^{\mu }$ such that $\left|\right\{\xi :|S_{\xi }\cap A|=\mu \left\}\right|<\lambda$ whenever $A\in {\left[\kappa \right]}^{<\kappa }$. We show that if $\Phi (\mu ,\kappa ,\lambda )$ holds and the space $X$ is both $\mu$-compact and $\lambda$-compact then $X$ is $\kappa$-compact as well. Moreover, from PCF theory we deduce $\Phi (cf\left(\kappa \right),\kappa ,{\kappa }^{+})$ for every singular cardinal $\kappa$. As a corollary we get that a linearly Lindelöf and ${\aleph }_{\omega }$-compact space is uncountably compact, that is $\kappa$-compact for all uncountable cardinals $\kappa$.

Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z)\ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of ${\ell }_{\infty }$.

The principle that "any product of cofinite topologies is compact" is equivalent (without appealing to the Axiom of Choice) to the Boolean Prime Ideal Theorem.

In this paper $LJ$-spaces are introduced and studied. They are a common generalization of Lindelöf spaces and $J$-spaces researched by E. Michael. A space $X$ is called an $LJ$-space if, whenever $\{A,B\}$ is a closed cover of $X$ with $A\cap B$ compact, then $A$ or $B$ is Lindelöf. Semi-strong $LJ$-spaces and strong $LJ$-spaces are also defined and investigated. It is demonstrated that the three spaces are different and have interesting properties and behaviors.

Nous nous proposons de rendre à Émile Borel le mérite d’avoir considéré le premier un recouvrement d’un segment de droite par une suite infinie d’intervalles et prouvé que l’on peut en extraire un sous-recouvrement fini. L’appellation de théorème de Heine-Borel souvent donnée à ce résultat, en référence à un article de Heine de 1872, conduit à sous-estimer les différences avec le théorème sur la continuité uniforme (dont une première version peut être attribuée à Dirichlet, en 1854) ; cette dénomination...