2015/12/30 by Maria Vincenta Ferrer, Maria Ferrer, Salvador Hernández +1 · 2 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Countable set #Geometric and Algebraic Topology #Group (periodic table) #Homomorphism #Normal subgroup #Pointwise #Pointwise convergence #Second-countable space #Upper and lower bounds #math.FA #math.GN #math.GR #msc:20C15 #msc:20K30 #msc:22A05 #msc:22A25 #msc:54B10 #msc:54D30 #msc:54H11
paper · pdf · doi:10.1090/proc/13532
published in Proceedings of the American Mathematical Society 145(8), 3599-3605 (American Mathematical Society)
arxiv created 2015/12/30 · openalex created_date 2016/06/24 · openalex publication_date 2017/02/23 · arxiv updated 2017/05/18 · openalex updated_date 2026/08/05
We prove that the group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals normal upper H normal o normal m left-parenthesis double-struck upper Z Superscript double-struck upper N Baseline comma double-struck upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">H</mml:mi> <mml:mi mathvariant="normal">o</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G=\mathrm Hom(\mathbb Z^\mathbb N, \mathbb Z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of all homomorphisms from the Baer-Specker group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Superscript double-struck upper N"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb Z^\mathbb N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that the second Pontryagin dual 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 discrete. As <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 non-discrete, it is not reflexive. Since <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> can be viewed as a closed subgroup of the Tychonoff product <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Superscript German c"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">c</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb Z^\mathfrak c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of continuum many copies of the integers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, this provides an example of a group described in the title, thereby resolving a problem by Galindo, Recoder-Núñez and Tkachenko. It follows that an inverse limit of finitely generated (torsion-)free discrete abelian groups need not be reflexive.