1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 105 E~ample The set Q of the rationals in IB.^1 with the relative metric is open and totally di ...

106 Chapter^2 . 14. If Ai and A 2 are starlike regions with respect to the same point z 0 , prove that both Ai U A 2 and Ai n A ...

Topology of Plane Sets of Points 107 The set UaEA Nv(a) is an open covering of A. Hence it contains a finite subcovering, say n ...

108 Chapter 2 Q = {(x, y): a::; x::; a+ l, b::; y::; b + l}, l > 0 being the side length of Q and (a, b) the bottom left vert ...

Topology of Plane Sets of Points then for any point ( x, y) E Q k we have d[(x, y), (a*, b*)] :::; d[(x, y), ( ak, bk)]+ d[( ak, ...

110 Chapter^2 is not empty. Hence the family of closed sets { G~ : a E I} has the finite intersection property. However, from LJ ...

Topology of Plane Sets of Points 111 metric 00 d(x, x'). = [2)xn - x~)^2 ] 1 f2 1 and let M be the subset M = {em: em = { 8~} ;: ...

112 Chapter 2 Notation If {xn}~ converges to x 0 , we write (as in r~al analysis) n-+oo lim {xn} = Xo or, briefly, n-+oo lim Xn ...

Topology of Plane Sets of Points 113 that d(xa,xi) 2: c and d(x 3 ,x 2 ) 2: c. Proceeding in the same manner, we obtain either a ...

114 Chapter^2 Fig. 2.rn Proof Suppose that n::'=l Fn = 0. This implies that U::'=i F~ = s. Be- cause S :is compact, it is the un ...

Topology of Plane Sets of Points 115 7. Let {Ga: a EI} be an open covering of the compact set ACC. Prove that there exists an c ...

116 Chapter 2 Examples (IR, d) is complete, whereas (Q, d) is not complete. Here d stands for the usual Euclidean metric. We ar ...

Topology of Plane Sets of Points 117 Proof 1. Necessity. Suppose that S is complete, and let En have center at an and radius rn, ...

118 Chapter 2 Theorem 2.28 (Cantor) A metric space '(s, d) is complete iff for any sequence {Fn} of nested nonempty closed sets ...

Topology of Plane Sets of Points 119 Definition 2.44 A complete metric space S* is said to be a completion of the space S iff (1 ...

120 Chapter 2 EXERCISES 2. 7 1. If (S, d) and (T, d') are complete metric spaces, show that the product space S x T with metric ...

Topology of Plane Sets of Points 121 The set S is called the space of the topology, and the collection of sets Q is called a top ...

122 Chapter^2 Let S = {a,b,c} and let n = {0, {a}, {b}, {a,b}, {a,b,c}}. Clearly ( S, n) is a topological space. However,. S is ...

Topology of Plane Sets of Points 123 Examples 1. Let S =IR and let a neighborhood of x 0 E IR be defined by N(x 0 ) = {x : x ;:: ...

124 Chapter^2 M. Frechet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), 1-74. P. R. Halmos, I ...