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The term is that's usually used is hereditarily equivalent.

As Will Brian pointed out, the pseudo-arc has this property. It is indecomposable, i.e., it is not the union of two of its proper subcontinua.

  G. W. Henderson showed that a hereditarily equivalent decomposable continuum is an arc, homeomorphic to $[0,1]$.

It is still a big open question as to whether the arc and the pseudo-arc are the only hereditarily equivalent metric continua.

Edit July 2020: The problem alluded to above has been solved for planar continua. The pseudo-arc is the only "minimal" (i.e. hereditarily equivalent) plane continuum other than the arc: https://doi.org/10.1016/j.aim.2020.107131.

The term that's usually used is hereditarily equivalent.

As Will Brian pointed out, the pseudo-arc has this property. It is indecomposable, i.e., it is not the union of two of its proper subcontinua. G. W. Henderson showed that a hereditarily equivalent decomposable continuum is an arc (homeomorphic to $[0,1]$).

It is still a big open question as to whether the arc and the pseudo-arc are the only hereditarily equivalent metric continua.

Edit July 2020: The problem alluded to above has been solved for planar continua. The pseudo-arc is the only "minimal" (i.e. hereditarily equivalent) plane continuum other than the arc: https://doi.org/10.1016/j.aim.2020.107131.

The term is that's usually used is hereditarily equivalent.

As Will Brian pointed out, the pseudo-arc has this property. It is indecomposable, i.e., it is not the union of two of its proper subcontinua.

G. W. Henderson showed that a hereditarily equivalent decomposable continuum is an arc, homeomorphic to $[0,1]$.

It is still a big open question as to whether the arc and the pseudo-arc are the only hereditarily equivalent metric continua.

The term is that's usually used is hereditarily equivalent.

As Will Brian pointed out, the pseudo-arc has this property. It is indecomposable, i.e., it is not the union of two of its proper subcontinua.

G. W. Henderson showed that a hereditarily equivalent decomposable continuum is an arc, homeomorphic to $[0,1]$.

It is still a big open question as to whether the arc and the pseudo-arc are the only hereditarily equivalent metric continua.

Edit July 2020: The problem alluded to above has been solved for planar continua. The pseudo-arc is the only "minimal" (i.e. hereditarily equivalent) plane continuum other than the arc: https://doi.org/10.1016/j.aim.2020.107131.

The term is that's usually used is hereditarily equivalent.

As Will Brian pointed out, the pseudo-arc has this property. It is indecomposable, i.e., it is not the union of two of its proper subcontinua. G. W. Henderson showed that a hereditarily equivalent decomposable continuum is an arc, homeomorphic to $[0,1]$.

The term is that's usually used is hereditarily equivalent.

As Will Brian pointed out, the pseudo-arc has this property. It is indecomposable, i.e., it is not the union of two of its proper subcontinua. 

G. W. Henderson showed that a hereditarily equivalent decomposable continuum is an arc, homeomorphic to $[0,1]$.

It is still a big open question as to whether the arc and the pseudo-arc are the only hereditarily equivalent metric continua.