Baire-Type Properties in Metrizable c0(Ω, X)

Ferrando and Lüdkovsky proved that for a non-empty set Ω and a normed space X, the normed space c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms .n∈N defining its topology, then c0(Ω,X) is the metrizable locally convex space over the field K (of the real or complex numbers) of all functions f:Ω→X such that for each ε>0 and n∈N the set ω∈Ω:f(ω)n>ε is finite or empty, with the topology defined by the semi-norms fn=supf(ω)n:ω∈Ω, n∈N. Kąkol, López-Pellicer and Moll-López also proved that the metrizable space c0(Ω,X) is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable c0(Ω,X) is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented.

Embeddability of right-angled Artin groups on complements of trees

For a finite simplicial graph [Formula: see text], let [Formula: see text] denote the right-angled Artin group on [Formula: see text]. Recently, Kim and Koberda introduced the extension graph [Formula: see text] for [Formula: see text], and established the Extension Graph Theorem: for finite simplicial graphs [Formula: see text] and [Formula: see text], if [Formula: see text] embeds into [Formula: see text] as an induced subgraph then [Formula: see text] embeds into [Formula: see text]. In this paper, we show that the converse of this theorem does not hold for the case [Formula: see text] is the complement of a tree and for the case [Formula: see text] is the complement of a path graph.

Non-archimedean closed graph theorems

We consider linear maps T: X ? Y, where X and Y are polar local convex spaces over a complete non-archimedean field K. Recall that X is called polarly barrelled, if each weakly* bounded subset in the dual X0 is equicontinuous. If in this definition weakly* bounded subset is replaced by weakly* bounded sequence or sequence weakly* converging to zero, then X is said to be l?-barrelled or c0-barrelled, respectively. For each of these classes of locally convex spaces (as well as the class of spaces with weakly* sequentially complete dual) as domain class, the maximum class of range spaces for a closed graph theorem to hold is characterized. As consequences of these results, we obtain non-archimedean versions of some classical closed graph theorems. The final section deals with the necessity of the above-named barrelledness-like properties in closed graph theorems. Among others, counterparts of the classical theorems of Mahowald and Kalton are proved.