Multivariable Advanced Calculus

10.2. DEFINITIONS AND ELEMENTARY PROPERTIES 259Proof: LetLet {k n } , {c n } be such thatd ≡ inf {||k − c|| : k ∈ K, c ∈ C}d + 1 n > ||k n − c n || .Since K is compact, there is a subsequence still denoted by {k n } such that k n → k ∈ K.Then also||c n − c m || ≤ ||c n − k n || + ||k n − k m || + ||c m − k m ||If d = 0, then as m, n → ∞ it follows ||c n − c m || → 0 and so {c n } is a Cauchy sequencewhich must converge to some c ∈ C. But then ||c − k|| = lim n→∞ ||c n − k n || = 0 and soc = k ∈ C ∩ K, a contradiction to these sets being disjoint. This proves the lemma. In particular the distance between a point and a closed set is always positive if thepoint is not in the closed set. Of course this is obvious even without the above lemma.10.2 Definitions And Elementary PropertiesIn this section, f : Ω → R n will be a continuous map. It is always assumed that f (∂Ω)misses the point y where d (f, Ω, y) is the topological degree which is being defined.Also, it is assumed Ω is a bounded open set.Definition 10.2.1 U y ≡ { f ∈ C ( Ω; R n) : y /∈ f (∂Ω) } . (Recall that ∂Ω = Ω \Ω) For two functions,f, g ∈ U y ,f ∼ g if there exists a continuous function,h : Ω × [0, 1] → R nsuch that h (x, 1) = g (x) and h (x, 0) = f (x) and x → h (x,t) ∈ U y for all t ∈[0, 1] (y /∈ h (∂Ω, t)). This function, h, is called a homotopy and f and g are homotopic.Definition 10.2.2 For W an open set in R n and g ∈ C 1 (W ; R n ) y is called aregular value of g if whenever x ∈ g −1 (y), det (Dg (x)) ≠ 0. Note that if g −1 (y) = ∅,it follows that y is a regular value from this definition. Denote by S g the set of singularvalues of g, those y such that det (Dg (x)) = 0 for some x ∈ g −1 (y).Lemma 10.2.3 The relation ∼ is an equivalence relation and, denoting by [f] theequivalence class determined by f, it follows that [f] is an open subset ofU y ≡ { f ∈ C ( Ω; R n) : y /∈ f (∂Ω) } .Furthermore, U y is an open set in C ( Ω; R n) and if f ∈ U y and ε > 0, there existsg ∈ [f] ∩ C 2 ( Ω; R n) for which y is a regular value of g and ||f − g|| < ε.Proof: In showing that ∼ is an equivalence relation, it is easy to verify that f ∼ fand that if f ∼ g, then g ∼ f. To verify the transitive property for an equivalencerelation, suppose f ∼ g and g ∼ k, with the homotopy for f and g, the function, h 1and the homotopy for g and k, the function h 2 . Thus h 1 (x,0) = f (x), h 1 (x,1) = g (x)and h 2 (x,0) = g (x), h 2 (x,1) = k (x). Then define a homotopy of f and k as follows.{h1 (x,2t) if t ∈ [ ]0, 1h (x,t) ≡ 2h 2 (x,2t − 1) if t ∈ [ 12 , 1] .