Almost completeness and the Effros open mapping principle in normed groups (Q2850622)

Consider a normed group \(G\), \(e=e_G\) the neutral element, and the right and a left norm topology via the right-invariant and left-invariant metrics \(d_R^T(s,t):=||st^{-1}||\) and \(d_L^T(s,t):=||s^{-1}t||\). It is a well-known property that the neighbourhoods of \(e_G\) under either norm-topology are the same. Now consider the following definitions: i) a normed group \(G\) acts continuously on \(X\) if there is a continuous mapping \(\phi:G\times X\longrightarrow X\) such that \(\phi(e,x)=x\) and \(\phi(gh,x)=\phi(g,\phi(h,x))\), ii) the action is transitive if for any \(x,y\) in \(X\) there is \(g\in G\) such that \(gx=y\), iii) the action is weakly micro-transitive if for each \(x\in X\) and each neighbourhood \(A\) of \(e_G\) the set cl\((Ax)=\) cl\(\{ax:a\in A\}\) has \(x\) as an interior point in \(X\), iv) the action is micro-transitive if for \(x\in X\) and each neighbourhood \(A\) of \(e_G\) the set \(Ax=\{ax:a\in A\}\) is a neighbourhood of \(x\).NEWLINENEWLINE The author makes an attempt to prove two versions of the famous Effros' theorem.NEWLINENEWLINEIn the process the concepts of analytic set, almost completeness and standard theorems like the analytic Cantor theorem and the characterization theorem for almost completeness are utilized.NEWLINENEWLINEThe changed versions of the Effros theorem are as follows : NEWLINENEWLINE(1) [Effros' theorem --- weak micro-transitive form] Let the normed group \(G\) act on \(X\) transitively. If \(G\) is separable under either norm topology and \(X\) is non-meagre, then the action of \(G\) is weakly micro-transitive. NEWLINENEWLINE(2) [Effros' theorem from weak micro-transitivity] If the normed group \(G\) is almost complete under the right-norm topology and the separately continuous action of \(G\) on \(X\) is weakly micro-transitive, then the action of \(G\) is micro-transitive.