In this course we will learn more advanced topics of general topology, and basic topics and examples in algebraic topology. Topology provides a general setting for studying continuous mathematics, and is a foundation for much of pure and applied mathematics. Algebraic topology translates difficult topological problems into computable algebraic questions. General topology topics include the Urysohn Lemma, Tietze Extension Theorem, Tychonoff theorem, Stone-\v Cech compactification. Algebraic topology topics include the fundamental group, the Seifert-van Kampen theorem, and the classification of surfaces. We will also learn some basic ideas of category theory.