Simultaneous Z/p-acyclic resolutions of expanding sequences

We prove the following Theorem: Let X be a nonempty compact metrizable space, let $l_1 \leq l_2 \leq…$beasequenceofnaturalnumbers,andlet$X_1 \subset X_2 \subset…$beasequenceofnonemptyclosedsubspacesofXsuchthatforeachkinN,$dim_{Z/p} X_k \leq l_k < \infty$. Then there exists a compact metrizable space Z, having closed subspaces $Z_1\subset Z_2\subset \mathrm{.}\mathrm{.}\mathrm{.}$, and a surjective cell-like map $\pi :Z\to X$, such that for each k in N, (a) $dim{Z}_k\le l_k$, (b) $\pi (Z_k)=X_k$, and © $\pi \mathrm{\mid }{Z}_k:Z_k\to X_k$ is a Z/p-acyclic map. Moreover, there is a sequence $A_1\subset A_2\subset \mathrm{.}\mathrm{.}\mathrm{.}$ of closed subspaces of Z, such that for each k, $dim{A}_k\le l_k$, $\pi \mathrm{\mid }{A}_k:A_k\to X$ is surjective, and for k in N, $Z_k\subset A_k$ and $\pi \mathrm{\mid }{A}_k:A_k\to X$ is a UV^{l_k-1}-map. It is not required that X be the union of all X_k, nor that Z be the union of all Z_k. This result generalizes the Z/p-resolution theorem of A. Dranishnikov, and runs parallel to a similar theorem of S. Ageev, R. Jim'enez, and L. Rubin, who studied the situation where the group was Z.