Universal Covers.
A cover of a space X is a space Y and a continuous surjection from Y to X such that the inverse image of a small neighborhood in X is a disjoint union of small neighborhoods in Y.
How is a cover related to a random variable, where inverse images of open sets are open?
If U is a universal cover of X and and Y is any other cover of X, then U can be made into a cover of Y in a natural way.
Variational Methods by Lawrence C. Evans.
Critical points.
One-dimensional variational problems.
Shortest distance in a plane is a straight line.