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29 March 2024
 
  » arxiv » 1202.4480

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On closed embeddings of free topological algebras
T.Banakh ; O.Hryniv ;
Date 20 Feb 2012
AbstractLet $mathcal K$ be a complete quasivariety of completely regular universal topological algebras of continuous signature $mathcal E$ (which means that $mathcal K$ is closed under taking subalgebras, Cartesian products, and includes all completely regular topological $mathcal E$-algebras algebraically isomorphic to members of $mathcal K$). For a topological space $X$ by $F(X)$ we denote the free universal $mathcal E$-algebra over $X$ in the class $mathcal K$. Using some extension properties of the Hartman-Mycielski construction we prove that for a closed subspace $X$ of a metrizable (more generally, stratifiable) space $Y$ the induced homomorphism $F(X) o F(Y)$ between the respective free universal algebras is a closed topological embedding. This generalizes one result of V.Uspenskii concerning embeddings of free topological groups.
Source arXiv, 1202.4480
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