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 = {X, p, A} is an inverse system of continuous images of arcs with monotone bonding mappings such that cf (card (A)) ≠ w, then X = lim X is a continuous image of an arc if and only if each proper subsystem {X, p, B} of X with cf(card (B)) = w has the limit which is a continuous image of an arc (Theorem 18).
In this paper we define a space σ() for approximate systems of compact spaces. The construction is due to H. Freudenthal for usual inverse sequences [4, p. 153–156]. We establish the following properties of this space: (1) The space σ() is a paracompact space, (2) Moreover, if is an approximate sequence of compact (metric) spaces, then σ() is a compact (metric) space (Lemma 2.4). We give the following applications of the space σ(): (3) If is an approximate system of continua, then X = lim is a...
For metrizable continua, there exists the well-known notion of a Whitney map. If is a nonempty, compact, and metric space, then any Whitney map for any closed subset of can be extended to a Whitney map for [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.
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