We say that a continuum $X$ is a hereditarily equivalent continuum (HEC) if every non-degenerate subcontinuum of it is homeomorphic to $X$. We can weaken this condition in three different levels: If considered in the hyperspace of continua of $X$, denoted by $\mathrm{Cont}(X)$, being hereditarily equivalent means that

This is an open and dense set, hence comeager, thus the first way to weaken it is to ask for the set of homeomorphic copies of $X$ to be a comeager subset of $\mathrm{Cont}(X)$. A continuum with this property we call a generically hereditarily equivalent continuum (GHEC). However, we can go further and consider the hyperspace of maximal order arcs $\mathrm{MOA}(X)$. In the case of an HEC, any maximal order arc is made of an initial unitary set called the root and homeomorphic copies of $X$, hence we can say that

• GCHEC holds for a space $X$ if comeager many elements of $\mathrm{MOA}(X)$ have this property of being a chain made of copies of $X$ apart from the root.
• GCGHEC holds for $X$ if comeager many elements of $\mathrm{MOA}(X)$ contain comeager many copies of $X$.

In this talk, we partially address two natural questions that arise from these definitions: “What kind of spaces satisfy these properties?” and “How are these properties related?”