Categorically closed countable semigroups
Categorically closed countable semigroups
Abstract In this paper, we establish a connection between categorical closedness and nontopologizability of semigroups. In particular, for the class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>𝖳</m:mi><m:mpadded lspace="-1.7pt" width="-1.7pt"><m:mn>𝟣</m:mn></m:mpadded></m:msub><m:mo></m:mo><m:mi>𝖲</m:mi></m:mrow></m:math> {\mathsf{T_{\!1}S}} of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>T</m:mi><m:mn>1</m:mn></m:msub></m:math> {T_{1}} topological semigroups we prove that a countable semigroup X with finite-to-one shifts is injectively <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>𝖳</m:mi><m:mpadded lspace="-1.7pt" width="-1.7pt"><m:mn>𝟣</m:mn></m:mpadded></m:msub><m:mo></m:mo><m:mi>𝖲</m:mi></m:mrow></m:math> {\mathsf{T_{\!1}S}} -closed if …