A map is continuous if the preimage of every open set is open. If the topology is defined by a metric, we can reformulate this as: the inverse image of an open ball contains an open ball . Like this:
But bringing these radii and into the picture will not serve any purpose unless we use them to quantify continuity. For example, if we insist that for a fixed constant , we arrive at the definition of a Lipschitz map.
But why do we look at the inverse image; what happens if we take the direct image instead? Then we get the definition of an open map: the image of every open set is open. Recasting this in metric terms: the image of an open ball contains an open ball . Like this:
If we quantify openness by requiring for a fixed , we arrive at the definition of a co-Lipschitz map. [Caution: some people use “co-Lipschitz” to mean , which is a different condition. They coincide if is bijective.]
I don’t know if openness without continuity is good for anything other than torturing students with exercises such as: “Construct an open discontinuous map from to .” We probably want both. At first one can hope that open continuous maps will have reasonable fibers : something -dimensional when going from dimensions to , with . The hope is futile: an open continuous map can squeeze a line segment to a point (construction left as an exercise).
A map that is both Lipschitz and co-Lipschitz is called a Lipschitz quotient; this is a quantitative analog of “open continuous”. It turns out that for any Lipschitz quotient the preimage of every point is a finite set. Moreover, factors as where is a complex polynomial and is a homeomorphism.
This is encouraging… but going even one dimension higher, it remains unknown whether a Lipschitz quotient must have discrete fibers. For an overview of the subject, see Bill Johnson’s slides.