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D.S. Lipham
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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.

D.S. Lipham
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