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

TL;DR

This paper characterizes one-to-one images of lines as composants in indecomposable continua, providing conditions for when such images are composants and demonstrating embeddings in Euclidean space.

Contribution

It establishes necessary and sufficient conditions for one-to-one line images to be composants of indecomposable continua and constructs embeddings in three-dimensional space.

Findings

Non-trivial one-to-one line images are indecomposable.

Conditions for a one-to-one image to be a composant involve dense mappings and convergent subsequences.

Existence of composant-preserving embeddings in Euclidean 3-space.

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.