The Shape and Cofinality of Generating Families

Let M be a subset of the real line. Then a real valued function f on M is discontinuous if and only if there is a sequence (x_n)_n on M converging to some x in M for which (f(x_n))_n does not converge to f(x). 

* What is the minimum size of a family of convergent sequences on M which witness discontinuity of every discontinuous function on M? 

Equivalently, what is the minimum size of a family of convergent sequences on M generating the topology of M? Applying ideas from the Tukey theory for relations, and Shelah's PCF theory we (largely) answer this question, and the related one on the minimum size of a family of compact subsets generating the topology.