\(\mathcal P\)-like spaces are limits of approximate \(\mathcal P\)-resolutions
\(\mathcal P\)-like metric compacta were defined and studied by the first author and the reviewer [Trans. Am. Math. Soc. 109, 146-164 (1963; Zbl 0118.390)], where it was shown that, for an arbitrary class \(\mathcal P\) of compact connected polyhedra, a metric compactum \(X\) is \(\mathcal P\)-like if and only if it is the limit of an inverse sequence of polyhedra from \(\mathcal P\) with surjective bonding mappings. It was proved by the first author [Period. Math.-Phys. Astron., II. Ser. 18, 195-205 (1963; Zbl 0144.219)] that the limit of an inverse system of compact polyhedra from \(\mathcal P\) (more generally, of compact \(\mathcal P\)-like spaces) with onto bonding mappings is always a \(\mathcal P\)-like compact space. However, for nonmetric \(\mathcal P\)-like compact spaces the converse need not hold. The first author and \textit{L. R. Rubin} [Pac. J. Math. 138, No. 1, 129-144 (1989; Zbl 0631.54006)] have introduced approximate inverse systems of metric compacta and their limits and have shown that compact Hausdorff spaces \(X\) with \(\dim X<n\) coincide with the limits of approximate systems of compact polyhedra of dimension \(< n\). The first author and the reviewer [Math. Jap. 33, No. 6, 895-908 (1988; Zbl 0668.54008)] have shown that for classes \(\mathcal P\) of compact polyhedra with no isolated points, \(\mathcal P\)-like compact Hausdorff spaces coincide with limits of approximate systems of polyhedra from \(\mathcal P\) with surjective bonding mappings. The purpose of this paper is to generalize the latter result to noncompact \(\mathcal P\)-like spaces and classes \(\mathcal P\) of locally compact polyhedra. The generalization uses the notion of an approximate resolution which was recently introduced by the first author and \textit{T. Watanabe} [Glas. Mat., III. Ser. 24(44), No. 4, 587-637 (1989; Zbl 0715.54009)] who combined the idea of an approximate inverse system of compacta with the idea of a resolution. The latter notion had been introduced by the first author [Fundam. Math. 114, 53-78 (1981; Zbl 0411.54019)] to replace the notion of inverse limit in noncompact situations.
scientific article; zbMATH DE number 90158
scientific article