Notes on Fréchet spaces

Abstract

Abstract. First, we introduce sequential convergence structures and characterize Fréchet spaces and continuous functions in Fréchet spaces using these structures. Second, we give sufficient conditions for the expansion of a topological space by the sequential closure operator to be a Fréchet space and also a sufficient condition for a simple expansion of a topological space to be Fréchet. Finally, we study on a sufficient condition that a sequential space be Fréchet, a weakly first countable space be first countable, and a symmetrizable space be semi-metrizable.

When a combination of convexity and continuity forces monotonicity of preferences

“…Consequently, any -continuous and convex preference relation on L 0 must violate the behavioral principle of payo¤-monotonicity. This implies that preferences which are induced by convex and monotone risk measures de…ned on the subdomain of loss random variables-as axiomatized, e.g., in Artzner et al (1997;1999) and Föllmer and Schied (2002)-cannot be -continuous.…”

“…Convexity is, e.g., satis…ed by coherent risk measures that have been axiomatized in order to address the non-convexity of the value-at-risk criterion (cf. Artzner et al 1997;1999;Delbaen 2002;2009). By part (iii) of Corollary 4, preferences that are induced by a convex risk measure : L 0 !…”

“…The family C forms a weak base for the C -topology so that C is weakly …rst-countable (or g-…rst countable) (cf. Arhangel'ski¼ ¬ 1966;Siwiec 1974;Hong 1999;Yang and Shi 2011). To see that any C 2 T C satis…es Condition BU, observe that there must be for every U 2 C with X 2 U some y U such that y 2 C (X) has the following form…”

The convergence-theoretic approach to g-first countable and symmetrizable spaces

“…The observation can be traced all the way back to [22], it was included without proof in [10], and a proof can be found in the popular survey [9]. Yet approaches to its proof have remained the subject of more research, e.g., [12,13] without yielding any refinement. The convergence-theoretic viewpoint provides as a consequence such a slight refinement by clarifying what condition is needed if separation is dropped, with an immediate and transparent proof.…”