  Given a compact geodesic space $X$ we apply the fundamental group and
alternatively the first homology group functor to the corresponding Rips or
\v{C}ech filtration of $X$ to obtain what we call a persistence. This paper
contains the theory describing such persistence: properties of the set of
critical points, their precise relationship to the size of holes, the structure
of persistence and the relationship between open and close, Rips and \v{C}ech
induced persistences. Amongst other results we prove that a Rips critical point
$c$ corresponds to an isometrically embedded circle of length $3c$, that a
homology persistence of a locally contractible space with coefficients in a
field encodes the lengths of the lexicographically smallest base and that Rips
and \v{C}ech induced persistences are isomorphic up to a factor $3/4$. The
theory describes geometric properties of the underlying space encoded and
extractable from persistence.
