Size theory

In mathematics, size theory studies the properties of topological spaces endowed with formula_1-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance between size pairs.
A survey of size theory can be found in 
History and applications.
The beginning of size theory is rooted in the concept of size function, introduced by Frosini. Size functions have been initially used as a mathematical tool for shape comparison in computer vision and pattern recognition.
An extension of the concept of size function to algebraic topology was made in "Size homotopy groups for computation of natural size distances", 
where size homotopy groups were introduced, together with the natural pseudodistance for formula_1-valued functions.
An extension to homology theory (the size functor) was introduced in 
The size homotopy group and the size functor are strictly related to the concept of persistent homology group
studied in persistent homology. It is worth to point out that the size function is the rank of the formula_3-th persistent homology group, while the relation between the persistent homology group 
and the size homotopy group is analogous to the one existing between homology groups and homotopy groups.
In size theory, size functions and size homotopy groups are seen as tools to compute lower bounds for the natural pseudodistance. 
Actually, the following link exists between the values taken by the size functions formula_4, formula_5 and the natural pseudodistance 
formula_6 between the size pairs formula_7
An analogous result holds for size homotopy group.
The attempt to generalize size theory and the concept of natural pseudodistance to norms that are different from the supremum norm has led to the study of other reparametrization invariant norms.