On the topologies over semigroups and groups, which are defined by families of deviations and norms (Q1390483)

Let \(X\) be a semigroup and let \(\tau\) be a topology on \(X\). We say that a topology is defined by a family of pseudometrics (deviations) \(\mathcal F\) if \(\{\{y\in X\mid f(x,y)\leq\epsilon \}\mid f\in \mathcal F, x\in X, \epsilon >0\}\) is a subbase of \(\tau\). A pseudometric \(f\) is called invariant from the left (or from the right) if \(f(x,y)=f(zx,zy)\) (or \(f(x,y)=f(xz,yz)\)) for all \(x,y,z\in X\). A family of pseudometrics \(\mathcal F\) is \(X\)-closed from the right if \(f_z\in \mathcal F\) for all \(f\in \mathcal F\) and all \(z\in X\) where \(f_z(x,y)=f(xz,yz)\) for all \(x,y\in X\). Assume that \(\tau\) is defined by a family \(\mathcal F\) of pseudometrics invariant from the left. Characterizations are given when \(X\) with \(\tau\) is a topological semigroup. E.g. it is true if all right translations are continuous, or if \(\mathcal F\) is \(X\)-closed from the right. If \(X\) is a group and \(N\) is a norm of \(X\) then define a pseudometric invariant from the left \(f_N(x,y)=N(y^{-1}x)\) and a pseudometric invariant from the right \(g_N(x,y)=N(xy^{-1})\). Characterizations are given when \(X\) with \(\tau\) is a topological group. E.g. it is true if there exists a family of norms \(\mathcal N\) on \(X\) such that the topology defined by \(\{f_N\mid N\in \mathcal N\}\), the topology defined by \(\{g_N\mid N\in \mathcal N\}\) and \(\tau\) coincide, or if all translations are \(\tau\)-continuous and for an arbitrary neighborhood \(U\) of the unit of \(X\) there exists a continuous norm \(N\) on \(X\) with \(N(X\setminus U)=\{1\}\).