### 1.3. Basic Topology

ICA
 Theorem 1.3.X: Triangle Inequality For all complex numbers z and w we have: |z + w| |z| + |w|

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 Definition 1.3.x: Disks, Open, and Closed Sets The set D(z0, r) is the open disk centered at z0 with radius r. In other words: D(z0, r) = {z C : |z - z0| < r } A set U is called open if for every z U there exists an r > 0 such that D(z, r) U. A set C is called closed if its complement comp(C) is open.

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 Example 1.3.X: Simple sets inC If c is a (complex) constant, then clearly the set {z = c} is a single point in the plane. Describe the sets { Re(z) = c} { Im(z) = c} { Arg(z) = c} { |z| = c} { z = } { r cis(t) = c} { z = r cis(t): r = c} { z = r cis(t): t = c}

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 Example 1.3.X: Disks and friends The set { |z - c| < r } is an open disk centered at c with radius r (explain). What are: { |z - c| > r } { |z - c| r } { r1 < |z - c| < r2 } { c1 < Arg(z) < c2 } { |z - 2z| < r}
circles, disks (inside/outside), annulus, lines, half-planes

Interactive Complex Analysis, ver. 1.0.0
(c) 2006-2007, Bert G. Wachsmuth