# Hereditarily equivalent

Let $X$ be a continuum. We say that that $X$ is hereditarily equivalent if $X$ is homeomorphic to each of its non-degenerate subcontinua.

# Properties

Hereditarily equivalent implies hereditarily self-like and subcontinuum-like

Pseudo-arc is hereditarily equivalent

Pseudo-arc is hereditarily indecomposable

Hereditarily equivalent and decomposable continuum is an arc

Hereditarily equivalent continuum is tree-like