I am kind of curious what topics to read to understand this concept more.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose I want to find a function [itex]f: A \rightarrow B[/itex], where if you look at the pre-image of any point [itex]b \in B[/itex], the size of the pre-image of b will be quite large. Essentially, I want to find functions that map from very large spaces, to very small spaces, with a pretty uniform distribution. In other words, I want to find functions so that if you take any point [itex]b \in B[/itex], you'll find that [itex]|f^{-1}(b)|[/itex] will be large (infinite hopefully), and pretty much the same regardless of which point you pick.

What are the "terms" that speak to the size of pre-images of points in a set under a function? What sort of area would I study to get a better grasp on functions like this, so that I might be able to "build one from scratch"? Topology?

My apologizes for this oddly worded question, I just do not know what are the mathematical terms that express the things I'm looking for. The only thing that comes to mind that I could study is the kernal. I can look up tons of properties about the kernal of a set that would be useful. But the kernal is the set of points that map specifically to the null point. What would be the concept that is like a kernal, but for points going to any particular point, not just the null point?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# What concept would I study, to be able to analyze the size of pre-images of sets under functions?

**Physics Forums | Science Articles, Homework Help, Discussion**