In this work, we prove three types of results with the strategy that,
together, the author believes these should imply the local version of Hilbert's
Fifth problem. In a separate development, we construct a nontrivial topology
for rings of map germs on Euclidean spaces. First, we develop a framework for
the theory of (local) nonstandard Lie groups and within that framework prove a
nonstandard result that implies that a family of local Lie groups that converge
in a pointwise sense must then differentiability converge, up to coordinate
change, to an analytic local Lie group, see corollary 6.1. The second result
essentially says that a pair of mappings that almost satisfy the properties
defining a local Lie group must have a local Lie group nearby, see proposition
7.1. Pairing the above two results, we get the principal standard consequence
of the above work, corollary 7.2, which can be roughly described as follows. If
we have pointwise equicontinuous family of mapping pairs (potential local
Euclidean topological group structures), pointwise approximating a (possibly
differentiably unbounded) family of differentiable (suffi- ciently approximate)
almost groups, then the original family has, after appropriate coordinate
change, a local Lie group as a limit point. The third set of results give
nonstandard renditions of equicontinuity criteria for families of
differentiable functions, see theorem 9.1. These results are critical in the
proofs of the principal results of this thesis as well as the standard
interpretations of the main results here. Following this material, we have a
long chapter constructing a Hausdorff topology on the ring of real valued map
germs on Euclidean space. This topology has good properties with respect to
convergence and composition. See the detailed introduction to this chapter for
the motivation and description of this topology
