754 CHAPTER 23. DEGREE THEORY
are restrictions of functions in Cc (Rp) , equivalently C (Rp) to Ω and by Cm(Ω),m ≤ ∞
the space of restrictions of functions in Cmc (Rp) , equivalently Cm (Rp) to Ω. If f ∈C
(Ω)
the symbol f will also be used to denote a function defined on Rp equalling f on Ω whenconvenient. The subscript c indicates that the functions have compact support. The normin C
(Ω)
is defined as follows.
∥ f∥∞,Ω = ∥ f∥
∞≡ sup
{| f (x)| : x ∈Ω
}.
If the functions take values in Rp write Cm(Ω;Rp
)or C
(Ω;Rp
)for these functions if there
is no differentiability assumed. The norm on C(Ω;Rp
)is defined in the same way as above,
∥f∥∞,Ω = ∥f∥
∞≡ sup
{|f(x)| : x ∈Ω
}.
Of course if m = ∞, the notation means that there are infinitely many derivatives. Also,C (Ω;Rp) consists of functions which are continuous on Ω that have values in Rp andCm (Ω;Rp) denotes the functions which have m continuous derivatives defined on Ω. Alsolet P consist of functions f(x) such that fk (x) is a polynomial, meaning an element of thealgebra of functions generated by
{1,x1, · · · ,xp
}. Thus a typical polynomial is of the form
∑i1···ip a(i1 · · · ip)xi1 · · ·xip where the i j are nonnegative integers and a(i1 · · · ip) is a realnumber.
Some of the theorems are simpler if you base them on the Weierstrass approximationtheorem.
Note that, by applying the Tietze extension theorem to the components of the function,one can always extend a function continuous on Ω to all of Rp so there is no loss of gener-ality in simply regarding functions continuous on Ω as restrictions of functions continuouson Rp. Next is the idea of a regular value.
Definition 23.0.2 For W an open set in Rp and g ∈C1 (W ;Rp) y is called a regular valueof g if whenever x ∈ g−1 (y), det(Dg(x)) ̸= 0. Note that if g−1 (y) = /0, it follows that y isa regular value from this definition. That is, y is a regular value if and only if
y /∈ g({x ∈W : detDg(x) = 0})
Denote by Sg the set of singular values of g, those y such that det(Dg(x)) = 0 for somex ∈ g−1 (y).
Also, ∂Ω will often be referred to. It is those points with the property that every openset (or open ball) containing the point contains points not in Ω and points in Ω. Then thefollowing simple lemma will be used frequently.
Lemma 23.0.3 Define ∂U to be those points x with the property that for every r > 0,B(x,r) contains points of U and points of UC. Then for U an open set,
∂U =U \U (23.0.1)
Let C be a closed subset of Rp and let K denote the set of components of Rp \C. Then ifK is one of these components, it is open and
∂K ⊆C