The number of topological ends
of the universal cover of a finite complex, K, is either 0, 1, 2, or ∞ and only depends
upon the fundamental group of K. Call this number e(K). We wish to define
numbers e(X) for compact metric spaces analogous to e(K). To accomplish this we
extend the theory of ends for finitely generated groups to certain inverse sequences of
finitely generated groups and their inverse limits. Classifications for these
inverse sequences and their inverse limits analogous to those for finitely
generated groups are derived. Whenever the fundamental pro-group of a
compact metric space, X, satisfies certain properties, we obtain a shape
invariant number e(X) (either 0, 1, 2 or ∞) and analyze what e(X) describes
geometrically.
