One-to-one composant mappings of $[0,\infty)$ and $(-\infty,\infty)$

Abstract

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 $[0,\infty)$ or $(-\infty,\infty)$, 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 $K(X)\cup \{X\}$. We also prove the existence of composant-preserving embeddings in Euclidean $3$-space. Accompanying the proofs are illustrations and examples.