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As Andreas points out in the comments, the pseudo-arcthe pseudo-arc provides an example of a continuum like this. This fact was first proved by Moise in 1948, in

Edwin Moise, "An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua," Transactions of the AMS 63 (1948), pp. 581-594.

As Andreas points out in the comments, the pseudo-arc provides an example of a continuum like this. This fact was first proved by Moise in 1948, in

Edwin Moise, "An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua," Transactions of the AMS 63 (1948), pp. 581-594.

As Andreas points out in the comments, the pseudo-arc provides an example of a continuum like this. This fact was first proved by Moise in 1948, in

Edwin Moise, "An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua," Transactions of the AMS 63 (1948), pp. 581-594.

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Will Brian
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As Andreas points out in the comments, the pseudo-arc provides an example of a continuum like this. This fact was first proved by Moise in 1948, in

Edwin Moise, "An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua," Transactions of the AMS 63 (1948), pp. 581-594.