Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces

Mine, Kotaro ; Sakai, Katsuro

Canadian journal of mathematics, Tome 63 (2011) no. 2, pp. 436-459

Résumé

Let $F$ be a non-separable $\text{LF}$ -space homeomorphic to the direct sum ${{\sum }_{n\in \text{N}}}\,{{\ell }_{2}}\left( {{\tau }_{n}} \right)$ , where ${{\aleph }_{0}}<{{\tau }_{1}}<{{\tau }_{2}}<\cdot \cdot \cdot $ . It is proved that every open subset $U$ of $F$ is homeomorphic to the product $\left| K \right|\,\times \,F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v\,\in \,{{K}^{\left( 0 \right)}}$ has the star $\text{St}\left( v,\,K \right)$ with card $\text{St}{{\left( v,K \right)}^{\left( 0 \right)}}<\tau =\sup {{\tau }_{n}}$ (and card ${{K}^{\left( 0 \right)}}\le \tau $ ), and, conversely, if $K$ is such a simplicial complex, then the product $\left| K \right|\,\times \,F$ can be embedded in $F$ as an open set, where $\left| K \right|$ is the polyhedron of $K$ with the metric topology.

Détail

DOI : 10.4153/CJM-2010-083-5

Mots-clés : 57N20, 46A13, 46T05, 57N17, 57Q05, 57Q40, LF-space, open set, simplicial complex, metric topology, locally finite-dimensional, star, small box product, ANR, l2(T), l2(T)-manifold, open embedding Σi∊N l 2(τi ).