Papers by Montserrat Bruguera
Leaning on a remarkable paper of Pryce, the paper studies two independent classes of topological ... more Leaning on a remarkable paper of Pryce, the paper studies two independent classes of topological Abelian groups which are strictly angelic when endowed with their Bohr topology. Some extensions are given of the Eberlein–ˇ theorem for the class of topological Abelian groups, and finally, for a large subclass of the latter, Bohr angelicity is related to the Schur property.
Sustainability, 2021
The COVID pandemic has touched many aspects of everyone’s life. Education is one of the fields gr... more The COVID pandemic has touched many aspects of everyone’s life. Education is one of the fields greatly affected by it, as students and teachers were forced to move online and quickly adapt to the online environment. Assessment is a crucial part of education, especially in STEM fields. A gap analysis was performed by expert groups in the frame of an Erasmus+ project looking at the practices of six European countries. Specialists teaching university-grade mathematics in seven European institutions were asked about their perception of gaps in the assessment of students both before (2019) and during (2021) the pandemic. This qualitative study looks at the difference in perception of such gaps after almost one year of online teaching. The analysis of their responses showed that some gaps were present before the pandemic, as well as others that are specific to it. Some gaps, such as the lack of IT infrastructure and the need to adapt materials to an online environment, have been exacerbat...
Topology and its Applications, 1996
Journal of the Australian Mathematical Society, 2003
We prove that in the character group of an abelian topological group, the topology associated (in... more We prove that in the character group of an abelian topological group, the topology associated (in a standard way) to the continuous convergence structure is the finest of all those which induce the topology of simple convergence on the corresponding equicontinuous subsets. If the starting group is furthermore metrizable (or even almost metrizable), we obtain that such a topology coincides with the compact-open topology. This result constitutes a generalization of the theorem of Banach-Dieudonné, which is well known in the theory of locally convex spaces.We also characterize completeness, in the class of locally quasi-convex metrizable groups, by means of a property which we have calledthe quasi-convex compactness property, or brieflyqcp(Section 3).
Journal of the London Mathematical Society, 2004
Leaning on a remarkable paper of Pryce, the paper studies two independent classes of topological ... more Leaning on a remarkable paper of Pryce, the paper studies two independent classes of topological Abelian groups which are strictly angelic when endowed with their Bohr topology. Some extensions are given of the Eberlein-Šmulyan theorem for the class of topological Abelian groups, and finally, for a large subclass of the latter, Bohr angelicity is related to the Schur property.
Topology and its Applications, 2014
We establish some general principles and find some counterexamples concerning the Pontryagin refl... more We establish some general principles and find some counterexamples concerning the Pontryagin reflexivity of precompact groups and P-groups. We prove in particular that: (1) A precompact Abelian group G of bounded order is reflexive iff the dual group G ∧ has no infinite compact subsets and every compact subset of G is contained in a compact subgroup of G. (2) Any extension of a reflexive P-group by another reflexive P-group is again reflexive. We show on the other hand that an extension of a compact group by a reflexive ωbounded group (even dual to a reflexive P-group) can fail to be reflexive. We also show that the P-modification of a reflexive σ-compact group can be nonreflexive (even if, as proved in [17], the P-modification of a locally compact Abelian group is always reflexive).
It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topolo... more It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topological vector spaces, to Abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to the Grothendieck theorem does not hold. By means of the continuous convergence structure on the dual of a topological group, we also state some weaker forms of the Grothendieck theorem valid for the class of locally quasi-convex groups. Finally, we prove that for the smaller class of nuclear groups, BB-reflexivity is equivalent to completeness.
We present a wide class of reflexive, precompact, non-compact, Abelian topological groups $G$ det... more We present a wide class of reflexive, precompact, non-compact, Abelian topological groups $G$ determined by three requirements. They must have the Baire property, satisfy the \textit{open refinement condition}, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group $G^\wedge$ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group $G$ of weight $\geq 2^\om$, we find proper dense subgroups $H_1$ and $H_2$ of $G$ such that $H_1$ is reflexive and pseudocompact, while $H_2$ is non-reflexive and almost metrizable.
We construct an Abelian topological group G and a closed subgroup N of G such that the groups N a... more We construct an Abelian topological group G and a closed subgroup N of G such that the groups N and G/N are countably compact, but G is not.
Houston journal of mathematics
Topology and its Applications, 2007
We show that the existence of a non-metrizable compact subspace of a topological group G often im... more We show that the existence of a non-metrizable compact subspace of a topological group G often implies that G contains an uncountable supersequence (a copy of the one-point compactification of an uncountable discrete space). The existence of uncountable supersequences in a topological group has a strong impact on bounded subsets of the group. For example, if a topological group G contains an uncountable supersequence and K is a closed bounded subset of G which does not contain uncountable supersequences, then any subset A of K is bounded in G \ (K \ A). We also show that every precompact Abelian topological group H can be embedded as a closed subgroup into a precompact Abelian topological group G such that H is bounded in G and all bounded subsets of the quotient group G/H are finite. This complements Ursul's result on closed embeddings of precompact groups to pseudocompact groups. (M. Bruguera), [email protected] (M. Tkachenko).
We show that the existence of a non-metrizable compact subspace of a topological group G often im... more We show that the existence of a non-metrizable compact subspace of a topological group G often implies that G contains an uncount- able supersequence (a copy of the one-point compactification of an uncountable discrete space). The existence of uncountable superse- quences in a topological group has a non-trivial impact on functionally bounded subsets of the group. For example, if a topological group G contains an uncountable supersequence and K is a closed func- tionally bounded subset of G which does not contain uncountable supersequences, then any subset A of K is functionally bounded in G n (K n A).
We study compact, countably compact, pseudocompact, and func- tionally bounded sets in extensions... more We study compact, countably compact, pseudocompact, and func- tionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G=N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G=N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample ex- ists under CH. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonne complete and "-spaces: one can find a pseudocompact, non-compact Abelian topo- logical group G and an infinite, closed, realcompact subgroup N of G 2000 Mathematics Subject Classification: Primary 54H11, 22A05; Secondary 54A20, 54G20.
Topology and its Applications, 1996
A number of attempts to extend Pontryagin duality theory to categories of groups larger than that... more A number of attempts to extend Pontryagin duality theory to categories of groups larger than that of locally compact abelian groups have been made using different approaches. The extension to the category of topological abelian groups created the concept of reflexive group. In this paper we deal with the extension of Pontryagin duality to the category of convergence abelian groups. Reflexivity in this category was defined and studied by E. Binz and H. Butzmann. A convergence group is reflexive (subsequently called BB-reflexive by us in our work) if the canonical embedding into the bidual is a convergence isomorphism.
Topology and its Applications, 2000
It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topolo... more It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topological vector spaces, to Abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to the Grothendieck theorem does not hold. By means of the continuous convergence structure on the dual of a topological group, we also state some weaker forms of the Grothendieck theorem valid for the class of locally quasi-convex groups. Finally, we prove that for the smaller class of nuclear groups, BB-reflexivity is equivalent to completeness. (M. Bruguera), [email protected] (M.J. Chasco), [email protected] (E. Martín-Peinador), [email protected] (V. Tarieladze).
Topology and its Applications, 2006
We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions o... more We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p = c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and µ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact 2000 Mathematics Subject Classification: Primary 54H11, 22A05; Secondary 54A20, 54G20.
Topology and its Applications, 1997
We prove in this paper that for a Hausdorff group topology on an Abelian group with sufficiently ... more We prove in this paper that for a Hausdorff group topology on an Abelian group with sufficiently many continuous characters, there is an associated locally quasi-convex topology which is the strongest among all the locally quasi-convex group topologies weaker than the given one. We a/so give a result on local quasi-convexity on the line of three-space properties. © 1997 Elsevier Science B.V.
Topology and its Applications, 2014
We establish some general principles and find some counter-examples concerning the Pontryagin ref... more We establish some general principles and find some counter-examples concerning the Pontryagin reflexivity of precompact groups and P -groups. We prove in particular that:
Uploads
Papers by Montserrat Bruguera