Chapter 1

Definition and Examples of Metric Spaces

Definition of a metric

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Euclidean and discrete metrics

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Metric on sequences and functions

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Sub-metric spaces

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Diameter of a set

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Chapter 2

Open, Closed Sets, and Convergence

Open balls and open sets

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Closed sets and limit points

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Interior, closure, and boundary

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Convergence of sequences in metric spaces

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Cauchy sequences in metric spaces

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Chapter 3

Completeness and Compactness

Complete metric spaces

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Cantor's intersection theorem

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Baire category theorem

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Compactness in metric spaces

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Heine-Borel theorem in R^n

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Chapter 4

Complex Numbers and Analytic Functions

Topology of the complex plane

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Functions of a complex variable

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Limits and continuity

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Differentiability of complex functions

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Analytic functions definition

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Chapter 5

Cauchy-Riemann Equations

Derivation of Cauchy-Riemann equations

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Sufficient conditions for differentiability

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Harmonic functions

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Orthogonal trajectories

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Construction of analytic functions

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Chapter 6

Contour Integration and Cauchy's Theorem

Contours and contour integrals

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Cauchy-Goursat theorem

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Cauchy integral formula

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Liouville's theorem

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Fundamental theorem of algebra

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