On topological Kadec norms (Q2487130)

For a separable Banach space \(({\mathbf B},|\cdot|)\) and a countable total collection \(A\) of linear functionals on \(B\), the \(A\)-topology on \(B\) is the weakest topology on \(B\) that makes every element of \(A\) continuous. The collection \(A\) is called norming if there is an equivalent norm \(\|\cdot\|\) on \(B\) that is lower semi-continuous (briefly, LSC) with respect to the \(A\)-topology. A new norm \(\|\cdot\|\) on \(B\) is a Kadec norm for \(A\) if it is topologically equivalent to \(|\cdot|\) and if the norm topology is the weakest topology on \(B\) that makes elements of \(A\) and the function \(\| \cdot\| \) continuous. It is well-known that \(A\) is norming for every Kadec norm for \(A\) and, conversely, the Kadec Renorming Theorem states that if \(A\) is norming, that \(A\) admits a Kadec norm. Evidently, for a topological space \(X_{\tau_1}=(X,\tau_1)\) and an LSC function \(f:X_{\tau_1}\to \mathbb{R}\), the weakest topology \(\tau_2\) on \(X\) that contains \(\tau_1\) and that makes \(f\) continuous, as a basis of open sets has the family \(\{\{x\in U: f(x)<t\}: U\in \tau_1\text{and }t\in \mathbb{R}\}\). In the present paper the authors consider only separable metric topologies and prove that for two separable metric topologies \({\mathcal W}\subset {\mathcal S}\) on the same set \(X\), every point in \(X\) has an \({\mathcal S}\)-neighbourhood basis consisting of \({\mathcal W}\)-closed sets if and only if there exists an LSC function \(\varphi:X_{\mathcal W}\to \mathbb{R}\) such that \({\mathcal S}\) is a weakest topology on \(X\) that contains \({\mathcal W}\) and that makes \(\varphi\) continuous (Theorem~1). An analogous situation is observed in the case of a fine topology in potential theory, studied, for example, by [\textit{M. Brelot}, On topologies and boundaries in potential theory (Springer-Verlag, Berlin-Heidelberg-New York) (1971; Zbl 0222.31014)]. From our point of view, Theorem~1 is a bitopological correspondence to the Kadec Renorming Theorem. Indeed, let \(B\) and \(A\) be as above; if \({\mathcal S}\) is the norm topology and if \({\mathcal W}\) is a \(A\)-topology, then on the one hand, the first condition of Theorem~1, which means exactly that the bitopological space \((X,{\mathcal W},{\mathcal S})\) is \(({\mathcal S},{\mathcal W})\)-regular in the sense of \textit{J. C. Kelly} [Proc. London Math. Soc., III. Ser. 13, 71--89 (1963; Zbl 0107.16401)], is equivalent to the statement that \(A\) is norming and, on the other hand, if \(\varphi\) is a norm \(\| \cdot\| \) on \(B\), then the second condition of Theorem~1 is satisfied if \(\| \cdot\| \) is a Kadec norm for \(A\). Theorem~1 has interesting corollaries connected with the notion of almost {\(n\)-dimensionality} and also with the different relations between the topological spaces \(({ X},{\mathcal W})\) and \(({ X},{\mathcal S})\).