# Math 434 syllabus

Topology

Course Description: An introduction to topology with emphasis on the geometric aspects of the subject. Topics covered include surfaces, topological spaces, open and closed sets, continuity, compactness, connectedness, product spaces, and identification and quotient spaces. Credit for both MA 434 and MA 542 is not allowed.
Prerequisites: MA 227 Minimum Grade of C and MA 237 Minimum Grade of C and MA 320 Minimum Grade of C.

Text: To be chosen by the instructor.

Learning outcomes: Upon the successful completion of the course a student will:

• Understand the definition of a topological space and related concepts, including subspaces, bases, separation axioms, density, and metrizability.
• Be familiar with important examples of topological spaces and surfaces, including discrete spaces, indiscrete spaces, Euclidean spaces and subspaces, the Sorgenfrey line, the annulus, the Möbius strip, the sphere, the torus, the Klein bottle, and the projective plane.
• Understand the definition and applications of continuous maps and homeomorphisms, including the Jordan curve theorem.
• Know various characterizations of compactness in terms of open covers and converging sequences.
• Recognize connected and path connected spaces, and know the counterexample of the topologist’s sine curve.
• Understand the product topology, and how properties such as metrizability and compactness are preserved by products.
• Know how points in topological spaces can be identified by way of equivalence relations and the quotient topology.
Improve their ability to discover and write mathematical proofs, as well as develop examples and counter-examples.