I assumed that for a point p in the sphere, there is a function F : p -> (f_1(p),...,f_k(p)). Hense the superscript k - that usually is the cardinality.

I have also have not seen such complicated use of the function arrow notation, e.g., . Usually you have just a specification of the domain and range, and the range is pretty simple, e.g., (just as an example). Afterwards one is given the arbitrarily complex function definition. In this instance, if I read your notation closely, it looks like the function is to take a sphere as input to an unspecified product of tuples as the output. Unspecified, in the sense that we do not yet have a product operator . I doubt that your intention was to say that the range of your function looks like a product of tuples (i.e., a complex structure); probably the range is the non-negative real numbers? Is it possible you're combining the range and domain from the arrow notation with elements of the function itself? Or is this a usage that is common that I just haven't seen before?