On group topologies and idempotents in weak almost periodic compactifications
Consider an abelian group G and group topologies \(\tau_ 1\) and \(\tau_ 2\). We write \(\tau_ 1\leq_ c\tau_ 2\) iff every \(\tau_ 2\) Cauchy filter is contained in a \(\tau_ 1\) Cauchy filter. This is tantamount to the existence of a compact subgroup K of \((G,\tau_ 1)\) such that the coset morphism \((G,\tau_ 2)\to (G,\tau_ 1)/K\) is continuous. If \(\tau_ 1\subseteq \tau_ 2\) then \(\tau_ 1\leq_ c\tau_ 2\). If \(\tau_ 1\leq_ c\tau_ 2\) and \(\tau_ 2\leq_ c\tau_ 1\) we write \(\tau_ 1=_ c\tau_ 2\). Let \(i_ G: G\to G^ w\) be the weakly almost periodic compactification. We denote by \({\mathcal T}\) the function from the lattice of idempotents \(E(G^ w)\) of the weak almost periodic compactification \(G^ w\) of G onto the set T(G) of weakly almost periodic (w.a.p.) topologies which associates with an idempotent e the pull-back topology under the morphisnm \(g\mapsto ei_ G(g): G\to eG^ w\). On the other hand let \({\mathcal E}\) denote the function \(T(G)\to E(G^ w)\) which assigns to a w.a.p. topology \(\tau\) the smallest idempotent in the kernel of the morphism \(G^ w\to (G_{\tau})^ w\). Theorem. For two w.a.p. topologies the relation \(\tau_ 1\leq_ c\tau_ 2\) holds iff \({\mathcal E}(t_ 1)\leq {\mathcal E}(t_ 2)\). The functions \({\mathcal E}\) and \({\mathcal T}\) implement a Galois connection between \(E(G^ w)\) and T(G) with \({\mathcal E}{\mathcal T}(e)=e\) for all e and \(\tau\) \(\subseteq {\mathcal T}{\mathcal E}(\tau)\) for all \(\tau\). In particular, \({\mathcal T}\) is injective and \({\mathcal E}\) is surjective.
scientific article; zbMATH DE number 4129111
scientific article