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We prove that the upper metric mean dimension of $C^0$-generichomeomorphisms, acting on a compact smooth boundaryless manifold with dimensiongreater than one, coincides with the dimension of the manifold. In the case ofcontinuous interval maps we also show that each level set for the metric meandimension is $C^0$-dense in the space of continuous endomorphisms of $[0,1]$with the uniform topology.

Generic homeomorphisms have full metric mean dimension