Characterizing local connectedness in inverse limits

Abstract

Let X denote the limit of an inverse system X -{X a p aa >; A} of locally connected Hausdorff continua. The main purpose of this paper is to define a notion of local connectedness for inverse systems, and to prove that if X is locally connected, then so is the limit X. If the bonding maps p aa > are surjections, then X is locally connected if and only if X is. The following corollaries are obtained. (1) If X is σ-directed and surjective, then X is locally connected. (2) If X is well-ordered, surjective, and weight (X«) ^ λ for each a in A, then either weight (X) ^ λ, or X is locally connected. (3) If X is σ -directed and the factor spaces X α are trees (generalized arcs), then X is a tree (generalized arc). (4) If X is well-ordered and the factor spaces X« are dendrites (arcs), then either X is metrizable, or X is a tree (generalized arc).1. Introduction. By a continuum we mean a compact connected Hausdorff space. Let X denote the limit of an inverse system X = {X α ; p αα A} where the factor spaces X a are locally connected continua, and A is an arbitrary directed set. It is well-known that every continuum X can be obtained as the limit of such a system where the factor spaces are polyhedra (see Theorem 10.1, p. 284, [2]). Hence local connectedness of the factor spaces X a does not imply local connectedness of the limit X. It is the main purpose of this paper to introduce a notion of local connectedness for inverse systems, and to prove that for such systems X the limit space X is locally connected (see Theorem 1). The converse holds if X is a surjective system, i.e., if the bonding maps p aa , are surjections. An immediate corollary is the known result that if X is a monotone inverse system, then X is locally connected [1].In §3 the main theorem is applied to well-ordered and σ-directed inverse systems, i.e., systems in which every countable subset of the index set is bounded above. The following somewhat surprising results are obtained. (1) If the inverse system X is ςr-directed and surjective, then the limit X is locally connected. (2) If X is well-ordered, surjective, and weight (X tt ) = λ for each a in A, then weight (X) Si λ or X is locally connected.Section 4 contains similar results about well-ordered and σ-directed inverse systems of trees (i.e., locally connected, hereditarily unicoherent continua [9]) and generalized arcs (i.e., ordered continua).

A Rim-Metrizable Continuum

“…Then X is a continuum as the inverse limit of continua. Since all the factor spaces Xa are locally connected continua and all the bonding maps ti are monotone and onto, X is a locally connected continuum (see, e.g., [4]). Thus we have the following properties of X : Claim 1.…”

A note on inverse limits of continuous images of arcs

“…This is impossible since x, y ∈ p Theorem 27 [3,Corollary 3]. Let X = {X a , p ab , A} be a σ-directed inverse system of hereditarily locally connected continua X a .…”

Combinatorial $${\mathbb{R}}$$ -trees as generalized Cayley graphs for fundamental groups of one-dimensional spaces

“…Let f k+1,∞ : X → X k+1 denote coordinate projection, i.e., f k+1,∞ ((x i ) i ) = x k+1 . By [15,Lemma 2], the numbers c k (v) are bounded by the (finite) number of (open) components of (f n • f n+1 • · · · • f k • f k+1,∞ ) −1 (Star(v, X n )) which intersect the (compact) set (f n • f n+1 • · · · • f k • f k+1,∞ ) −1 ({v}). Hence, the essential multiplicity of v is finite.…”

“…It suffices to show that X is locally connected. Following [15], we show that X has Property S: every open cover of X can be refined by a finite cover of connected subsets of X. Fix n and let {C 1 , C 2 , .…”

“…Let f k+1,∞ : X → X k+1 denote coordinate projection, i.e., f k+1,∞ ((x i ) i ) = x k+1 . By [15,Lemma 2], the numbers c k (v) are bounded by the (finite) number of (open) components of (…”