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

Published online by Cambridge University Press:  20 November 2018

Kotaro Mine  and
Katsuro Sakai
Kotaro Mine
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan email: [email protected]@sakura.cc.tsukuba.ac.jp
Katsuro Sakai
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan email: [email protected]@sakura.cc.tsukuba.ac.jp
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Abstract

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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.

Keywords

57N2046A1346T0557N1757Q0557Q40LF-spaceopen setsimplicial complexmetric topologylocally finite-dimensionalstarsmall box productANR, ℓ2(𝒯)ℓ2(𝒯)-manifoldopen embedding Σi∊ℕ𝓵2(τi).
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 2 , 01 April 2011 , pp. 436 - 459
Copyright
Copyright © Canadian Mathematical Society 2012

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