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We show that the set S(R) of shift-isomorphism classes of semidualizing complexes over a local ring R admits a nontrivial metric. We investigate the interplay between the metric and several algebraic operations. Motivated by the dagger duality isometry, we prove the following: If K,L are homologically bounded below and degreewise finite R-complexes such that K⊗_R^LK⊗_R^LL is semidualizing, then K is shift-isomorphic to R. In investigating the existence of nontrivial open balls in S(R), we prove that S(R) contains elements that are not comparable in the reflexivity ordering if and only if it contains at least three distinct elements.