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# 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
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## 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, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$
 MSC Classifications: 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

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