MT A531     General Topology I

MWF 10:00 - 10:50,  Room: 109 


 
Office hours:    MWF 1:00  - 2:00 Other hours by appointment.    
Office:    Ritter Hall 115        
Email:    barta@slu.edu  
Phone: (314)-977-2852

Texts:    J.R. Munkres, Topology: A First Course, 2/e
L.A. Steen and J.A. Seebach Jr., Counterexamples in Topology, Dover Publications.

Prerequisite:    MT A422 (Metric Spaces)

Grade: Homework, participation [20%], 2 Exams (25% each), Final (30%)

General topology, or point-set topology, defines and studies properties of spaces and maps such as connectedness, compactness and continuity.
Algebraic topology uses structures from abstract algebra, especially the group to study topological spaces and the maps between them. (http://en.wikipedia.org/wiki/Topology)

Topologies you will learn about in this course include - but are not limited to - the following list.
Compact-Open Topology, Discrete Topology, Finite Complement Topology, Indiscrete Topology, Lower Limit Topology, Metric Topology, Order Topology, Product Topology, Quotient Topology, Subspace Topology, Usual topology on the real line, Zariski topology (classical)

The topics from Munkres we will cover in the first semester are (tentatively):

Chapter 2
§12 Topological Spaces (pdf outline): Know the definition of a topology and understand the examples.

§13 Basis for a Topology (pdf outline): Know the definition of a basis and understand how a basis generates a topology.

§14 The Order Topology (pdf outline): This is one example of a topology. The order topology on the real line is the same as the usual topology. Get more interesting topologies if we look at the plane with the dictionary order.

§15 The Product Topology on X x Y (pdf outline): Basic definitions of the topologies on a product space. Understand the examples. More detail will be given in section 19. Here we only consider the fairly straight forward product of two spaces.

§16 The Subspace Topology (pdf outline). A subspace "inherits" the toplogy of the space it is contained in. The open sets in a subset A of X are just the intersection of the open sets in X with A. In other words we obtain the open sets of A by restricting the open sets of X to the subspace under consideration.

§17 Closed Sets and Limit Points (pdf outline): Definitions of closed sets and limit points. Hausdorff spaces are introduced.

§18 Continuous Functions (pdf outline)

§19 The Product Topology (versus Box Topology) (pdf outline)
§20 The Metric Topology (pdf outline)
§21 The Metric Topology (continued) (pdf outline)
§22 The Quotient Topology (pdf outline)
Supplementary Exercises: Topological Groups (pdf outline)

Chapter 3 Connectedness and Compactness
§23 Connected Spaces
§24 Connected Subspaces of the Real Line
§25 Components and Local Connectedness
§26 Compact Spaces
§27 Compact Subspaces of the Real Line
§28 Limit Point Compactness
§29 Local Compactness.

Chapter 4 Countability and Separation Axioms
§30 The Countability Axioms
§31 The Separation Axioms.
§32 Normal Spaces
§33 The Urysohn Lemma
§34 The Urisohn Metrization Theorem
(skip 35 and 36)

Chapter 5 The Tychonoff Theorem
§37 The Tychonoff Theorem
§38 The Stone-Cech Compactification
 
(Skip Chapter 6)

Chapter 7 Complete Metric Spaces and Function Spaces
§43 Complete Metric Spaces
(skip 44)
§45 Compactness in Metric Spaces
§46 Pointwise and Compact Convergence
§47 Ascoli’s Theorem

Chapter 9 The Fundamental Group.
§51 Homotopy of Paths
§52 Fundamental Groups