Let
$A$
be the inductive limit of a system
$${{A}_{1\,}}\,\xrightarrow{{{\phi }_{1,\,2}}}\,{{A}_{2}}\,\xrightarrow{{{\phi }_{2,\,3}}}\,{{A}_{3}}\,\to \cdots $$
with
${{A}_{n}}\,=\,\oplus _{i=1}^{{{t}_{n}}}\,{{P}_{n,\,i}}{{M}_{\left[ n,\,i \right]}}(C({{X}_{n,\,i}})){{P}_{n,\,i}}$
, where
${{X}_{n,\,i}}$
is a finite simplicial complex, and
${{P}_{n,\,i}}$
is a projection in
${{M}_{[n,i]}}\,\left( C\left( {{X}_{n,i}} \right) \right)$
. In this paper, we will prove that
$A$
can be written as another inductive limit
$${{B}_{1}}\,\xrightarrow{{{\psi }_{1,\,2}}}\,{{B}_{2}}\,\xrightarrow{{{\psi }_{2,\,3}}}\,{{B}_{3}}\,\to \cdots $$
with
${{B}_{n}}\,=\,\oplus _{i=1}^{{{s}_{n}}}\,{{Q}_{n,i}}{{M}_{\left\{ n,\,i \right\}}}(C({{Y}_{n,\,i}})){{Q}_{n,\,i}}$
, where
${{Y}_{n,\,i}}$
is a finite simplicial complex, and
${{Q}_{n,\,i}}$
is a projection in
${{M}_{\left\{ n,\,i \right\}}}(C({{Y}_{n,\,i}}))$
, with the extra condition that all the maps
${{\psi }_{n,n+1}}$
are injective. (The result is trivial if one allows the spaces
${{Y}_{n,\,i}}$
to be arbitrary compact metrizable spaces.) This result is important for the classification of simple
$\text{AH}$
algebras. The special case that the spaces
${{X}_{n,\,i}}$
are graphs is due to the third author.