Cell structures and completely metrizable spaces and their mappings

Wojciech Dębski   ; E. D. Tymchatyn ¹

¹ Department of Mathematics and Statistics University of Saskatchewan 106 Wiggins Rd. Saskatoon, SK S7N 5E6, Canada

Colloquium Mathematicum, Tome 147 (2017) no. 2, pp. 181-194

Résumé

A (combinatorial) graph is a discrete set of vertices together with a set of edges. We define cell structures as inverse sequences of graphs with mild convergence conditions and we define cell mappings between cell structures. These cell structures yield completely metrizable spaces as perfect images of closed subsets of countable products of discrete spaces. Cell mappings between cell structures define the continuous mappings between the corresponding spaces. In this way we can envision a continuous mapping between metric spaces as the limit of a sequence of discrete approximations. Thus, cell structures provide a kind of bridge between discrete and continuous mathematics.

Détail

DOI : 10.4064/cm6576-10-2016

Keywords: combinatorial graph discrete set vertices together set edges define cell structures inverse sequences graphs mild convergence conditions define cell mappings between cell structures these cell structures yield completely metrizable spaces perfect images closed subsets countable products discrete spaces cell mappings between cell structures define continuous mappings between corresponding spaces envision continuous mapping between metric spaces limit sequence discrete approximations cell structures provide kind bridge between discrete continuous mathematics