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 $\operatorname{Cont}(X)$, being hereditarily equivalent means that

$\operatorname{Cont}(X)\setminus {{x} \ \ x \in X} = { K \in \operatorname{Cont}(X) \ \ K \simeq X}.$

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 $\operatorname{Cont}(X)$. 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 $\operatorname{MOA}(X)$. 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 $\operatorname{MOA}(X)$ 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 $\operatorname{MOA}(X)$ 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?”