2009/09/30 by Dikran Dikranjan, Gábor Lukács · 10 citations
Mathematics · #Advanced Topology and Set Theory #Closure (psychology) #Combinatorics #Component (thermodynamics) #Connected component #Curse of dimensionality #Dimension (graph theory) #Group (periodic table) #Homotopy and Cohomology in Algebraic Topology #Mathematics #Physics #Pure mathematics #Rings, Modules, and Algebras #Statistics #Topological group #Topology (electrical circuits) #Zero (linguistics) #math.GN #math.GR #msc:22A05 #msc:22D05 #msc:54D05 #msc:54D25 #msc:54H11
paper · pdf · doi:10.1090/s0002-9939-2011-10626-9
published in Proceedings of the American Mathematical Society 139(8), 2995-3008 (American Mathematical Society) · Paper was completely rewritten since v2, and several new results (Theorems B, C, and D) were added
arxiv created 2010/02/15 · openalex publication_date 2011/03/23 · arxiv updated 2011/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A topological group is <italic>locally pseudocompact</italic> if it contains a non-empty open set with pseudocompact closure. In this paper, we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a group with the property that every closed subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is locally pseudocompact, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 0"> <mml:semantics> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">G0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is dense in the component of the completion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper G 0"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">G/G0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is zero-dimensional. We also provide examples of hereditarily disconnected pseudocompact groups with strong minimality properties of arbitrarily large dimension, and thus show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G slash upper G 0"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">G/G0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> may fail to be zero-dimensional even for totally minimal pseudocompact groups.