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Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces |
Canad. J. Math. 63(2011), 436-459
Published:2010-12-29
Printed: Apr 2011
Printed: Apr 2011
- Kotaro Mine,
- Katsuro Sakai,
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 |