2017/07/17 by David S. Lipham, David Lipham, Lipham, David
Mathematics · #54A20 #54C10 #54F15 #54F50 #FOS: Mathematics #General Topology (math.GN) #Rings, Modules, and Algebras #math.GN #msc:54A20 #msc:54C10 #msc:54F15 #msc:54F50
paper · pdf · doi:10.48550/arxiv.1707.05007
12 pages, 5 figures
openalex publication_date 2017/07/17 · arxiv created 2018/06/24 · arxiv updated 2018/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Knaster continua and solenoids are well-known examples of indecomposable continua whose composants (maximal arcwise-connected subsets) are one-to-one images of lines. We show that essentially all non-trivial one-to-one composant images of (half-)lines are indecomposable. And if f is a one-to-one mapping of [0,∞) or (-∞,∞), then there is an indecomposable continuum of which X:=ran(f) is a composant if and only if f maps all final or initial segments densely and every non-closed sequence of arcs in X has a convergent subsequence in the hyperspace K(X)∪ \X\. We also prove the existence of composant-preserving embeddings in Euclidean 3-space. Accompanying the proofs are illustrations and examples.