Stability of Persistence Modules

We present a new proof of the algebraic stability theorem, perhaps the main theorem in the theory of stability of persistent homology. We also give an example showing that an analogous result does not hold for a certain class of $\mathbb{R}^2$-modules. Persistent homology is a method in applied topology used to reveal the structure of certain types of data sets, e.g. point clouds in $\mathbb{R}^n$, by computing the homology of a parametrized set of topological spaces associated to the data set. Results like the algebraic stability theorem give a theoretical justification for the use of persistence homology in practice by showing that a small amount of noise in the input only influences the output by a similarly small amount