Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.2160

Minimality in topological groups and Heisenberg type groups

Menachem Shlossberg
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study relatively minimal subgroups in topological groups. We find, in particular, some natural relatively minimal subgroups in unipotent groups which are defined over "good" rings. By "good" rings we mean archimedean absolute valued (not necessarily associative) division rings. Some of the classical rings which we consider besides the field of reals are the ring of quaternions and the ring of octonions. This way we generalize in part a previous result which was obtained by Dikranjan and Megrelishvili and involved the Heisenberg group.

Explore related subjects

math.GNmath.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Menachem Shlossberg. 2007-06-14. Minimality in topological groups and Heisenberg type groups. https://arxiv.org/abs/0706.2160

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hemicompactness and the $k_ω$-property of free topological vector spaces

For a Tychonoff space $X$ we consider the vector space $E(X)$ with basis $X$ endowed with one of the following topologies: the weak topologies $L_b(X)$ and $L_p(X)$ generated by $C_b(X)$ and $C(X)$, the free locally convex space $L(X)$, the free locally $r$-convex space $L^{(r)}(X)$ ($0<r\le1$), the free locally pseudoconvex space $L^{(0)}(X)$, and the free topological vector space $V(X)$. We prove that $L_b(X)$ is hemicompact if and only if $X$ is finite; that for every vector topology between $L_p(X)$ and $L^{(0)}(X)$ hemicompactness is equivalent to $X$ being countable with all compact subsets finite; and that $V(X)$, as well as the free topological groups $F(X)$ and $A(X)$, are hemicompact if and only if $X$ is hemicompact. In particular, $V([0,1])$ is hemicompact, while $L([0,1])$ and $L_p([0,1])$ are not. We also describe when these spaces are $k_ω$-spaces: $L_b(X)$ and $L_p(X)$ only for finite $X$, and $L^{(r)}(X)$, $0\le r\le1$, only for countable discrete $X$.

math.GN↗
arXiv2026

$PF$-well-filtered spaces

This paper solves two open problems concerning the well-filteredness of topological spaces. The first, due to Shen, Xi and Zhao\cite{shen2023further}, asks whether a space must be open well-filtered whenever its upper space is so; the answer is negative; the Hofmann-Lawson space provides a counterexample. The second, posed by Li, Yuan and Zhao\cite{li2020unified}, inquires whether $PF$-well-filteredness and $PF$- sobriety are equivalent under the assumption of local compactness; we answer it in the affirmative. Subsequently, we give characterizations of $PF$-sobriety and $PF$-well-filteredness for $T_0$ spaces, respectively. On the basis of the characterizations, we obtain the following result: $PF$-sobriety and $PF$-well-filteredness coincide on the Rudin spaces, which extends the previously established equivalence for locally compact spaces, as every locally compact space is a Rudin space. Furthermore, we give a counterexample to show that for nonempty $T_0$ topological spaces $X$ and $Y$, if $Y$ is a $PF$-well-filtered ($PF$-sober) space, then the function space consisting of all continuous maps from $X$ to $Y$ equipped with the Isbell topology may not be a $PF$-well-filtered ($PF$-sober) space even when $X$ is a core-compact space. We also prove that if the function space comprising all continuous maps from nonempty $X$ to $Y$ endowed with the Isbell topology is $PF$-well-filtered ($PF$-sober), then $Y$ is also $PF$-well-filtered ($PF$-sober) even though $PF$-sobriety and $PF$-well-filteredness are not preserved under retracts. Finally, we show that the reflective hull of the category of $PF$-well-filtered ($PF$-sober) spaces with continuous maps in $TOP_0$ is $TOP_0$.

math.GN↗
arXiv2026

A non-sober core-compact meet-continuous dcpo

We construct a meet-continuous dcpo whose Scott space is core-compact but neither sober nor locally compact. The example is a proper sub-dcpo of a locally compact sober complete lattice, and restriction induces an isomorphism between their lattices of Scott-closed sets. It follows that a locally compact sober dcpo need not be determined, among all dcpos, by its Scott-closed-set lattice.

math.GN↗