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We say that a metric space (X, d) possesses the Banach Fixed Point Property (BFPP) if every contraction f : X → X has a fixed point. The Banach Fixed Point Theorem states that every complete metric space has the BFPP. However, E. Behrends pointed out in 2006 that the converse implication does not hold; that is, the BFPP does not imply completeness; in particular, there is a nonclosed subset of R 2 possessing the BFPP. He also asked if there is even an open example in R n , and whether there isdoi:10.1090/s0002-9939-09-09904-3 fatcat:skj3u4jwwnbbtahsz5qaxtsbuy