Abstract view
Simplicial Complexes and Open Subsets of NonSeparable LFSpaces


Published:20101229
Printed: Apr 2011
Kotaro Mine,
Institute of Mathematics, University of Tsukuba, Tsukuba, 3058571, Japan
Katsuro Sakai,
Institute of Mathematics, University of Tsukuba, Tsukuba, 3058571, Japan
Abstract
Let $F$ be a nonseparable LFspace 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 finitedimensional 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: 
LFspace, open set, simplicial complex, metric topology, locally finitedimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$
LFspace, open set, simplicial complex, metric topology, locally finitedimensional, 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 infinitedimensional manifolds [See also 58Bxx] Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40] Infinitedimensional manifolds [See also 53Axx, 57N20, 58Bxx, 58Dxx] Topology of topological vector spaces General topology of complexes Regular neighborhoods
57N20  Topology of infinitedimensional manifolds [See also 58Bxx] 46A13  Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40] 46T05  Infinitedimensional manifolds [See also 53Axx, 57N20, 58Bxx, 58Dxx] 57N17  Topology of topological vector spaces 57Q05  General topology of complexes 57Q40  Regular neighborhoods
