### ${\aleph}_{1}$-directed inverse systems of continuous images of arcs.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

The main purpose of this paper is to prove some theorems concerning inverse systems and limits of continuous images of arcs. In particular, we shall prove that if X = {Xa, pab, A} is an inverse system of continuous images of arcs with monotone bonding mappings such that cf (card (A)) ≠ w1, then X = lim X is a continuous image of an arc if and only if each proper subsystem {Xa, pab, B} of X with cf(card (B)) = w1 has the limit which is a continuous image of an arc (Theorem 18).

Short proofs of the fact that the limit space of a non-gauged approximate system of non-empty compact uniform spaces is non-empty and of two related results are given.

We prove that a map between two realcompact spaces is skeletal if and only if it is homeomorphic to the limit map of a skeletal morphism between ω-spectra with surjective limit projections.

An approximate inverse sequence of plane continua is constructed which negatively answers a question of S. Mardeši’c related to approximate and usual inverse systems. The example also shows that an important result of M.G. Charalambous cannot be improved. As an application, it is shown that a procedure of making an approximate inverse sequence commutative (“taming”) is discontinuous.

The fundamental properties of approximate inverse systems of uniform spaces are established. The limit space of an approximate inverse sequence of complete metric spaces is the limit of an inverse sequence of some of these spaces. This has an application to the dimension of the limit space of an approximate inverse system. A topologically complete space with $dim\le n$ is the limit of an approximate inverse system of metric polyhedra of $dim\le n$. A completely metrizable separable space with $dim\le n$ is the limit of an...

Realcompact spaces can be characterized as limits of approximate inverse systems of Polish polyhedra.

The main result of the present paper is a classification theorem for finite-sheeted covering mappings over connected paracompact spaces. This theorem is a generalization of the classical classification theorem for covering mappings over a connected locally pathwise connected semi-locally 1-connected space in the finite-sheeted case. To achieve the result we use the classification theorem for overlay structures which was recently proved by S. Mardesic and V. Matijevic (Theorems 1 and 4 of [5]).

In the existing literature there are several constructions of the strong shape category of topological spaces. In the one due to Yu. T. Lisitsa and S. Mardešić [LM1-3] an essential role is played by coherent polyhedral (ANR) expansions of spaces. Such expansions always exist, because every space admits a polyhedral resolution, resolutions are strong expansions and strong expansions are always coherent. The purpose of this paper is to prove that conversely, every coherent polyhedral (ANR) expansion...