  In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence
modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism
classes of) persistence modules, the interleaving distance. The definitions of
$\epsilon$-interleavings and $d_I$ generalize readily to multidimensional
persistence modules. In this paper, we develop the theory of multidimensional
interleavings, with a view towards applications to topological data analysis.
We present four main results. First, we show that on 1-D persistence modules,
$d_I$ is equal to the bottleneck distance $d_B$. This result, which first
appeared in an earlier preprint of this paper, has since appeared in several
other places, and is now known as the isometry theorem. Second, we present a
characterization of the $\epsilon$-interleaving relation on multidimensional
persistence modules. This expresses transparently the sense in which two
$\epsilon$-interleaved modules are algebraically similar. Third, using this
characterization, we show that when we define our persistence modules over a
prime field, $d_I$ satisfies a universality property. This universality result
is the central result of the paper. It says that $d_I$ satisfies a stability
property generalizing one which $d_B$ is known to satisfy, and that in
addition, if $d$ is any other pseudometric on multidimensional persistence
modules satisfying the same stability property, then $d\leq d_I$. We also show
that a variant of this universality result holds for $d_B$, over arbitrary
fields. Finally, we show that $d_I$ restricts to a metric on isomorphism
classes of finitely presented multidimensional persistence modules.
