# Continuous curves of nonmetric pseudo-arcs and semi-conjugacies to interval maps

## TL;DR

This paper extends the construction of nonmetric pseudo-arcs to new continua like pseudo-circles and solenoids, and shows that any interval map can be semi-conjugated to a homeomorphism of such nonmetric pseudo-arcs.

## Contribution

It introduces new nonmetric continua, including pseudo-circles and solenoids, and demonstrates semi-conjugacy of interval maps to homeomorphisms of nonmetric pseudo-arcs.

## Key findings

- Constructed a nonmetric pseudo-circle and nonmetric solenoids.
- Proved semi-conjugacy of interval maps to nonmetric pseudo-arc homeomorphisms.
- Extended previous metric pseudo-arc results to nonmetric contexts.

## Abstract

In 1985 M. Smith constructed a nonmetric pseudo-arc; i.e. a Hausdorff homogeneous, hereditary equivalent and hereditary indecomposable continuum. Taking advantage of a decomposition theorem of W. Lewis, he obtained it as a long inverse limit of metric pseudo-arcs with monotone bonding maps. Extending his approach, and the results of Lewis on lifting homeomorphisms, we construct a nonmetric pseudo-circle, and new examples of homogeneous 1-dimensional continua; e.g. a circle and solenoids of nonmetric pseudo-arcs. Among many corollaries we also obtain an analogue of another theorem of Lewis from 1984: any interval map is semi-conjugate to a homeomorphism of the nonmetric pseudo-arc.