Defining topologies on semigroups with invariant measure
Let X be a semigroup and let S be a ring of subsets of X. Suppose that \(A\in S\) and \(x\in X\) imply \(xA=\{xt:\) \(t\in A\}\) belongs to S. Let \(\mu\) be an additive nonnegative left translation invariant function on S. Let \(f_ A\) be the function on \(X\times X\) such that \(f_ A(s,t)=\mu(sA\Delta tA)\) for all s and \(t\in X\). Let \(S_ 0\) be a nonempty subset of S such that \(0<\mu(A)<+\infty\) for all \(A\in S_ 0\). The Weyl topology \(\tau_ W\) is the uniform topology obtained by taking as base all sets of the form \(\{x\in X:\quad f_{A_ i}(x,y)<\epsilon,\quad i=1,2,...,n\},\) where \(A_ i\in S_ 0\), \(\epsilon>0\). In this paper the author proves the following theorems. Theorem 1: If \(A\in S_ 0\) and \(a\in K\) imply \(aA\in S_ 0\), then the mapping \((x,y)\mapsto xy\) of \(X\times X\) into X is continuous. Moreover, if X is an invertible semigroup, then the mapping \(x\mapsto x^{-1}\) of X into X is continuous. Particularly, if X is a group, then \((X,\tau_ W)\) is a topological group. Theorem 2: If for any x,\(y\in X\), \(x\neq y\) there exists a set \(A\in S_ 0\) such that \(f_ A(x,y)>0\) and \(S_ 0\) contains a countable dense subset of X, then \((X,\tau_ W)\) is a metrizable topological space. Theorem 3: Let \((X,\tau)\) be a topological semigroup. If \(\sup p(\mu)=X\) and \((\cup_{t\in U}\rho_ t^{-1}(yU))_{U\in \tau}\) is a base of neighborhoods of y for all \(y\in X\), where \(\rho_ a:x\mapsto xa\), then the Weyl topology \(\tau_ W\) on X coincides with \(\tau\).
scientific article; zbMATH DE number 3855395