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.