Spinas, Otmar Linear topologies on sesquilinear spaces of uncountable dimension. (English) Zbl 0766.03027 Fundam. Math. 139, No. 2, 119-132 (1991). Summary: F. Appenzeller [J. Symb. Logic 54, No. 3, 689-699 (1989; Zbl 0687.03030)] introduced an invariant \(\Gamma\) for orthosymmetric sesquilinear spaces of regular uncountable dimension \(\kappa\) which takes its values in some Boolean algebra \({\mathcal D}(\kappa)\). Constructively he shows that \(\Gamma\) maps onto \({\mathcal D}(\omega_ 1)\). We show that this is not true for \(\kappa>\omega_ 1\). Orthosymmetric sesquilinear spaces naturally bear linear topologies defined by the form. There are various relations between the arithmetic, geometric and topological properties of such spaces. E.g., W. Bäni [Commun. Algebra 5, 1561-1587 (1977; Zbl 0365.15002)] characterizes \(\gamma\)-diagonal spaces using the notions of \(\gamma\)-compactness and continuous bases. We present an alternative characterization: existence of a convergent algebraic basis. Bäni [loc. cit.] asked whether there exist \(\gamma\)-compact spaces without continuous bases for arbitrary regular \(\gamma\). We give a positive answer by showing that the spaces defined by Appenzeller are examples of this. Cited in 1 ReviewCited in 2 Documents MSC: 03E35 Consistency and independence results 15A63 Quadratic and bilinear forms, inner products 46A19 Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.) Keywords:regular cardinal; club-filter; closed unbounded set; orthosymmetric sesquilinear spaces; regular uncountable dimension; linear topologies; \(\gamma\)-diagonal spaces; convergent algebraic basis; \(\gamma\)-compact spaces; continuous bases Citations:Zbl 0687.03030; Zbl 0365.15002 PDFBibTeX XMLCite \textit{O. Spinas}, Fundam. Math. 139, No. 2, 119--132 (1991; Zbl 0766.03027) Full Text: DOI EuDML