Natural pseudodistance
In size theory, the natural pseudodistance between two size pairs ( M , φ : M → R ) {\displaystyle (M,\varphi :M\to \mathbb {R} )\ } , ( N , ψ : N → R ) {\displaystyle (N,\psi :N\to \mathbb {R} )\ } is the value inf h ‖ φ − ψ ∘ h ‖ ∞ {\displaystyle \inf _{h}\|\varphi -\psi \circ h\|_{\infty }\ } , where h {\displaystyle h\ } varies in the set of all homeomorphisms from the manifold M {\displaystyle M\ } to the manifold N {\displaystyle N\ } and ‖ ⋅ ‖ ∞ {\displaystyle \|\cdot \|_{\infty }\ } is the supremum norm. If M {\displaystyle M\ } and N {\displaystyle N\ } are not homeomorphic, then the natural pseudodistance is defined to be ∞ {\displaystyle \infty \ } .