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Natural pseudodistance

In size theory, the natural pseudodistance between two size pairs , is the value , where varies in the set of all homeomorphisms from the manifold to the manifold and is the supremum norm. If and are not homeomorphic, then the natural pseudodistance is defined to be . It is usually assumed that , are closed manifolds and the measuring functions are . Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from to .
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