arXiv subjects

Tayomara Borsich

Publications and source records attributed to Tayomara Borsich.

3 recordsLinked to original sources

arXiv2026

On continuous homomorphisms from Alexandroff paratopological groups into topological groups

Alexandroff paratopological groups provide a natural setting in which order-theoretic and algebraic properties interact. In this short note we prove obstruction results when considering continuous homomorphisms from Alexandroff paratopological groups into topological groups. Particularly and, as a consequence of these results, we provide an alternative proof of the fact that the only connected Alexandroff topological group is the one that carries the trivial topology.

math.GN↗

arXiv2026

On the existence and properties of Alexandroff paratopological groups

We study groups endowed with Alexandroff topologies and show that no non-discrete Alexandroff topology can turn a group into a topological group. This settles negatively the basic existence problem for Alexandroff topological groups. Motivated by this obstruction, we turn to the broader setting of Alexandroff paratopological groups. We establish several fundamental properties of these spaces and provide explicit non-compact $T_0$ examples, showing that the Alexandroff framework is rich enough to capture nontrivial paratopological phenomena. As applications, we address two classical open questions concerning feebly bounded subsets in paratopological groups, proving that non-compact Alexandroff paratopological groups offer a positive solution both for products of feebly bounded sets and for the feebly boundedness of $B^2$ when $B$ is a feebly bounded subset.

math.GR↗

arXiv2021

On the existence of topologies compatible with a group duality with predetermined properties

The paper deals with group dualities. A group duality is simply a pair $(G, H)$ where $G$ is an abstract abelian group and $H$ a subgroup of characters defined on $G$. A group topology $τ$ defined on $G$ is {\it compatible} with the group duality (also called dual pair) $(G, H)$ if $G$ equipped with $τ$ has dual group $H$. A topological group $(G, τ)$ gives rise to the natural duality $(G, G^\wedge)$, where $G^\wedge$ stands for the group of continuous characters on $G$. We prove that the existence of a $g$-barrelled topology on $G$ compatible with the dual pair $(G, G^\wedge)$ is equivalent to the semireflexivity in Pontryagin's sense of the group $G^\wedge$ endowed with the pointwise convergence topology $σ(G^\wedge, G)$. We also deal with $k$-group topologies. We prove that the existence of $k$-group topologies on $G$ compatible with the duality $(G, G^\wedge)$ is determined by a sort of completeness property of its Bohr topology $σ(G, G^\wedge)$ (Theorem 3.3).

math.GN↗