Hereditarily indecomposable continuum

Knaster continuum

An indecomposable continuum in which every proper subcontinuum is also indecomposable.

Comments

A famous example is the pseudo-arc, see [a1] and [a2].

References

[a1]	E.E. Moise, "An indecomposable plane continuum which is homeomorphic to each of its non-degenerate subcontinua" Trans. Amer. Math. Soc. , 63 (1948) pp. 581–594
[a2]	R.H. Bing, "A homogeneous indecomposable plane continuum" Duke Math. J. , 15 (1948) pp. 729–742