  The main aim of this paper is to explore the ideas of persistent homology and
extended persistent homology, and their stability theorems, using ideas from
[Bubenik and Scott, 2014; Cohen-Steiner, Edelsbrunner, and Harer, 2007; and
Cohen-Steiner, Edelsbrunner, and Harer, 2009], as well as other sources. The
secondary aim is to explore the homology (and cohomology) of non-orientable
surfaces, using the Klein bottle as an example. We also use the Klein bottle as
an example for the computation of (extended) persistent homology, referring to
it throughout the paper.
