Generalized cell maps
Carlos Islas, Benjamín A. Itzá Ortiz, Rocío Leonel
TL;DR
The paper extends the notion of cell maps to generalized cell structures (g-cell structures), enabling a broader class of spaces to be realized as quotients of inverse limits. It defines weak g-cell maps and g-cell maps, analyzes when maps between inverse limits induce continuous maps on quotient spaces $G^{igstar}$ and $H^{igstar}$, and provides both existence results and continuity criteria. A key contribution is showing how cell maps induce weak g-cell maps and how g-cell maps yield corresponding induced maps on quotients, along with constructive methods to realize a given continuous $F:G^{igstar} o H^{igstar}$ as $ighat{f}$ under additional hypotheses. The work also clarifies limitations by presenting examples where continuity fails to lift from quotients or from set-valued maps, thereby mapping the boundary between discrete and non-discrete settings in generalized cellular frameworks. Overall, this extends representability and continuity transfer in inverse-limit constructions to a wider, non-discrete context with practical implications for topology and related areas.
Abstract
In [6] the notion of a g-cell structure was introduced as a generalization of the construction proposed by Debski and Tymchatyn to realize a certain class of topological spaces as quotient spaces of inverse limits. In [2], cell maps are defined between cell structures and showed that a cell map between two complete cell structures induces a continuous function between the spaces determined by the cell structures. In this paper, for g-cell structures, we introduce the notions of weak g-cell maps and g-cell maps. We give some conditions for weak g-cell and g-cell mappings between g-cell structures to induce continuous mappings between their corresponding spaces.