In size theory, the natural pseudodistance between two size pairs , is the value , where varies in the set of all homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

s 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 manifold

Closed manifold

In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....

s 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 .

The concept of natural pseudodistance can be easily extended to size pairs where the measuring function takes values in
.

Main properties

It can be proved
that the natural pseudodistance always equals the Euclidean distance between two critical values of the measuring functions (possibly, of the same measuring function) divided by a suitable positive integer .
If and are surfaces, the number can be assumed to be , or . If and are curves, the number can be assumed to be or .
If an optimal homeomorphism exists (i.e., ), then can be assumed to be .

See also

• Fréchet distance
  Fréchet surface
  In mathematics, a Fréchet surface is an equivalence class of parametrized surfaces in a metric space. In other words, a Fréchet surface is a way of thinking about surfaces independently of how they are "written down"...
  
  
• Size theory
• Size function
  Size function
  Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane xSize functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane...
  
  
• Size functor
  Size functor
  Given a size pair \ where M\ is a manifold of dimensionn\ and f\ is an arbitrary real continuous function definedon it, the i\ -th size functor, with i=0,\ldots,n\ , denoted...
  
  
• Size homotopy group
• Size pair