 
Summary: Submanifolds.
Let n be a positive integer.
Definition. We say f is an ndiffeomorphism if
(1) f is function whose domain and range of f are open subsets of Rn
;
(2) f is smooth;
(3) f is univalent and, for each x dmn f, f(x) carries Rn
isomorphically onto itself.
Whenever U and V are open subsets of Rn
we let
Diffeon
be the set of ordered triples (U, F, V ) such that F is an ndiffeomorphism with domain U and range V .
Proposition. We have
(1) is an ndiffeormorphism;
(2) if F is an ndiffeormorphism and W is an open subset of Rn
then FW is an ndiffeormorphism;
(3) if U is a family of open subsets of Rn
, F : U Rn
, F is univalent and FU is a ndiffeormorphism
for each U U then F is an ndiffeormorphism;
