On iterated inverse limits
by
Sibe Mardešić
University of Zagreb

Let X be the limit of a (cofinite) inverse system X = (X_\lambda, p_{\lambda\lambda'}, \Lambda) of compact Hausdorff spaces X_\lambda and let every X_\lambda be the limit of an inverse system Y_\lambda = (Y_\lambda^\mu, q_\lambda^\mu\mu', M_\lambda) of compact ANR's (polyhedra). During the Dubrovnik 1998 Topology Conference Yu.T. Lisica asked if one can organize the collection {Y_\lambda^\mu, \lambda in \Lambda, \mu in M_\lambda} of ANR's in such a way that one obtains an inverse system whose limit is X.

Lisica's question has a negative answer. This can be seen by taking a compact Hausdorff space X with dim X=1 and ind X=2 and a system X of metric compacta of dimesion 1 with limX = X and by expressing each X_\lambda as the limit of a sequence of 1-dimensional compact polyhedra. Nevertheless, for every cofinite system X there exist systems X_\lambda of compact ANR's with limit X_\lambda, whose terms can be organized in a system with limit X. Using resolutions instead of limits, this positive result generalizes to arbitrary topological spaces X.

References

S. Mardesi\'c and N. Uglesi\'c: On iterated inverse limits, Topology and its Appl. (to appear).

Date received: June 9, 2000