1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

1. 6 • T HE TOPOLOGY OF COMPLEX NUMBERS 41

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Figure L24 An e neighborhood of the point zo,

• EXAMPLE L24 The solut ion sets of the inequalities lzl < 1, lz -ii < 2,
  and lz + 1 + 2il < 3 are neighborhoods of the points 0, i , and - 1 - 2i , with
  radii 1, 2, and 3 , respectively, They can also be expressed as D 1 (0), D 2 (i), and
  DJ (- 1 - 2i),

We also define D, (zo), the closed dis k of radius e centered at zo, and
D; (zo), the punctured dis k of r adius e centered at zo, as

D, (zo) = {z: lz - zol::; e} , and

D; (zo) = {z: 0 < lz -zol < e}.

(1-50)

(1-51)

T he point zo is said to be an interior point of the set S provided that
there exists an e neighborhood of zo that contains only points of S; zo is called
an exterior point of the set S if there exists an e neighborhood of zo that
contains no points of S , If zo is neither an interior point nor an exterior point
of S, then it is called a boundary point of Sand has the property that each e
neighborhood of zo contains both points in S and points not in S, Figure L25
illustrates this situation.

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Figure l,2 5 The interior, exterior, and boundary of a set.