## 1.3. Basic Topology | ICA |

Theorem 1.3.X: Triangle InequalityFor all complex numbers zandwwe have:|z + w| |z| + |w|

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Definition 1.3.x: Disks, Open, and Closed SetsThe set D(zis the open disk centered at_{0}, r)zwith radius_{0}r. In other words:A setD(z_{0}, r) = {z: |z - zC_{0}| < r }is calledUopenif for everyzthere exists anUr > 0such thatD(z, r). A setUis calledCclosedif its complementcomp(is open.)C

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Example 1.3.X: Simple sets inCIf cis 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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circles, disks (inside/outside), annulus, lines, half-planes

Example 1.3.X: Disks and friendsThe set { |z - c| < r }is an open disk centered atcwith radiusr(explain). What are:

{ |z - c| > r }{ |z - c| r }{ r_{1}< |z - c| < r_{2}}{ c_{1}< Arg(z) < c_{2}}{ |z - 2z| < r}