CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

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

  Published:2010-12-29
 Printed: Apr 2011
  • Kotaro Mine,
    Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan
  • Katsuro Sakai,
    Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF  

Abstract

Let $F$ be a non-separable LF-space homeomorphic to the direct sum $\sum_{n\in\mathbb{N}} \ell_2(\tau_n)$, where $\aleph_0 < \tau_1 < \tau_2 < \cdots$. It is proved that every open subset $U$ of $F$ is homeomorphic to the product $|K| \times F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v \in K^{(0)}$ has the star $\operatorname{St}(v,K)$ with $\operatorname{card} \operatorname{St}(v,K)^{(0)} < \tau = \sup\tau_n$ (and $\operatorname{card} K^{(0)} \le \tau$), and, conversely, if $K$ is such a simplicial complex, then the product $|K| \times F$ can be embedded in $F$ as an open set, where $|K|$ is the polyhedron of $K$ with the metric topology.
Keywords: LF-space, open set, simplicial complex, metric topology, locally finite-dimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$-manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$ LF-space, open set, simplicial complex, metric topology, locally finite-dimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$-manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$
MSC Classifications: 57N20, 46A13, 46T05, 57N17, 57Q05, 57Q40 show english descriptions Topology of infinite-dimensional manifolds [See also 58Bxx]
Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40]
Infinite-dimensional manifolds [See also 53Axx, 57N20, 58Bxx, 58Dxx]
Topology of topological vector spaces
General topology of complexes
Regular neighborhoods
57N20 - Topology of infinite-dimensional manifolds [See also 58Bxx]
46A13 - Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40]
46T05 - Infinite-dimensional manifolds [See also 53Axx, 57N20, 58Bxx, 58Dxx]
57N17 - Topology of topological vector spaces
57Q05 - General topology of complexes
57Q40 - Regular neighborhoods
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/