Minimal flows with a closed proximal cell (Q2730702)

For a flow \((X,T)\) , where \(X\) is a compact Hausdorff space and \(T\) a discrete group on \(X\), points \(x\) and \(y\) are proximal if every neighborhood \(W\) of the diagonal \(\triangle\) in \(X\times Y\) there is \(t\in T\) such that \((xt,yt)\in W\). Points \(x\) and \(y\) are regionally proximal if for every neighborhood \(U\) of \(x\), \(V\) of \(y\) and \(W\) of \(\triangle\) there are \(x'{\i}U\), \(y'\in V\) and \(t\in T\) such that \((x't,y't)\in W\). The proximal cell of \(x\) is the set \(P(x)=[y\in X; (x,y)\in P],\) where \(P\) denotes the proximal relation. It is shown that regional proximality is an equivalence relation in a minimal flow containing a closed proximal cell. Further an equivalent conditioon to a local Bronstein condition is given.