LECTURE 5. MULTI-VALUED PERTURBATIONS 217closed sets with empty interior.^3 The set ex(M -Mreg) is the union of these images
over all i, and hence is also a countable union of closed sets with empty interior.
Hence the set IR - ex(M - Mreg) is of the second category in the sense of Baire, i.e.
is a countable intersection of dense open sets. Hence the set of all regular values of
ex in the complement of ex(M - Mreg) is also of the second category in the sense ofBaire and, in particular, it is dense. This shows that there exists an Ex > 0 which
satisfies the first two conditions above and that Ex can be chosen arbitrarily small.
Next observe that if x E M(j) - Mi for some i E J(j) then x ~ cl(Mi) and hence
infM; ex > 0. This shows that a sufficiently small number Ex > 0 satisfies the third
condition as well. Thus we have proved the existence of a constant Ex > 0 which
satisfies the above requirements.
Now define
Ux = {y EM : ex(Y) <Ex}.
Then Ux is an open neighbourhood of x and it is equal to the union of the branches
Min Ux over all i E J(j) with x E Mi. Moreover, since Ex ~ ex(M - Mreg) is a
regular value of the map ex o l.{Ji-^1 : l.{Ji(Mi) -t [O, 1] for every i E J(j), the set
l.{Ji(Mi n Ux ) = { t E l.{Ji(Mi) : ex 0 i.{Ji -l (t) <Ex}
is a finite union of open intervals with boundary points in l.{Ji(Mreg)· Now cover M
by finitely many such sets Ux. This proves Step l.
Step 2: There exists a finite set of vertices V C Mreg U 8M, a finite collection of
continuous injections la : [O, 1] -t M, indexed by a E A, and a map A -t I: a~
i(a) such that the following holds.
• For every a the endpoints xo = la(O) and xi = la(l) lie in the set V and
la([O, 1]) n V = {xo,x1}.
Moreover, i.{Ji(a) o la is an orientation preserving diffeomorphism from [O, 1]
onto the interval [to, t1], where to= i.{Ji(a)(xo) and ti= i.{Ji(a)(x1).
- For every x E M - V,
.A(x) = L Ai(a)·
xE1 0 ([0,l])• IfxEV-8M,
l o (O)=x l o(l)=x3If 1 : M -t N is a smooth map between manifolds of the same dimension and A C M is a
compact set with empty interior, then l(A) C N is a compact set with empty interior. Since
A is compact we may assume without loss of generality that the topology of M has a countable
basis and so Sard's theorem applies. Now suppose, by contradiction, that int(f(A)) f= 0. By
Sard's theorem, the function 1 has a regular value y E int(f (A)). Since A is compact, 1-^1 (y) n A
is a finite set, and we denote 1-^1 (y) n A = { x1, ... , Xm}. Choose compact neighbourhoods Ui
of Xi such that, for every i, the restriction of 1 to Ui is a diffeomorphism onto some compact
neighbourhood of y. Then there exists a compact neighbourhood V C l(A) of y such that, for
every x E A with 1 ( x) E V, there exists an i E { 1, ... , m} such that x E Ui. Otherwise there
would be a sequence x.., E A -LJi Ui such that l(x..,) converges toy and then any limit point
of x.., would be a preimage of y in A not equal to any of the Xi· This contradiction shows that
the neighbourhood V exists. By definition, this neighbourhood satisfies V C LJi l(Ui n A). But
l(Ui nA) is a compact set with empty interior for every i. By Baire's category theorem, V cannot
be the union of finitely many such sets. This contradiction shows that l(A) has empty interior.