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Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces
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- Journal:
- Canadian Journal of Mathematics / Volume 63 / Issue 2 / 01 April 2011
- Published online by Cambridge University Press:
- 20 November 2018, pp. 436-459
- Print publication:
- 01 April 2011
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- Article
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- You have access
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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.
