Rigorous foundations of point set topology: topological spaces, continuous maps. Metric spaces. Connectedness, compactness, countability and separation axioms. Product topology, quotient topology. Introduction to algebraic topology: fundamental group, the notion of a covering space, van Kampen theorem, computations and applications. The course provides foundations for graduate study in topology, geometry, and analysis and is recommended for advanced math majors. Students seeking an introductory elective in topology should enroll in MAT 364.