## Course 212 (Topology) in the Academic Year 2000-01

Course 212 (Topology), Academic Year 2000-01, Part I
Topics covered included limits and continuity, open and closed sets in Euclidean spaces and metric spaces.
Course 212 (Topology), Academic Year 2000-01, Part II
Topics covered topological spaces, including in particular product topologies, identification maps and quotient topologies, compactness and connectedness.
Course 212 (Topology), Academic Year 2000-01, Part III
Topics covered included normed vector spaces.
Course 212 (Topology), Academic Year 2000-01, Part IV
Topics covered included the exponential map defined on the complex plane and winding numbers, with applications to topology in the plane.

These notes document Course 121 (Topology) as it was taught in the academic years 1998-99, 1999-2000 and 2000-2001.

## Course 212 (Topology) in the Academic Year 1998-99

Course 212 (Topology), Academic Year 1998-99, Problems

Problem sets I and II were also distributed, with small changes, in the academic years 1999-2000 and 2000-01.

## Course 212 (Topology) in the Academic Year 1991-92

Course Notes:—

Course 212 (Topology), Academic Year 1991—92, Section 1: Metric Spaces
This section introduces and develops the theory of metric spaces
Course 212 (Topology), Academic Year 1991—92, Section 2: Topological Spaces
This section introduces and develops the theory of topological spaces
Course 212 (Topology), Academic Year 1991—92, Section 3: Connected Topological Spaces
This section proves various results concerning the topological notions of connectedness and path-connectedness.
Course 212 (Topology), Academic Year 1989—90, Section 4: Compact Topological Spaces
This section proves various results concerning the topological notion of compactness.
Course 212 (Topology), Academic Year 1989—90, Section 5: Complete and Compact Metric Spaces
This section introduces the concept of a complete metric spaces and presents the construction of the completion of a metric space, then continues to prove a number of results specific to compact metric spaces.
Course 212 (Topology), Academic Year 1989—90, Section 6: Normed Vector Spaces
This section introduces the concept of a smooth manifold and applies theorems concerning compactness in order to prove results concerning the existence of partitions of unity subordinate to open covers on smooth manifolds.
Course 212 (Topology), Academic Year 1989—90, Section 7: Introduction to Functional Analysis
This section introduces homotopies that represent continuous deformations of continuous maps, and introduce the definition of the fundamental group of a topological space.
Course 212 (Topology), Academic Year 1989—90, Section 8: The Exponential Map
This section describes the theory of covering maps
Course 212 (Topology), Academic Year 1989—90, Section 9: Winding Numbers
This section develops the theory of winding numbers of closed curves in the complex plane

Problems:—

Course 212, Problems 1991-92

## Course 212 (Topology) in the Academic Year 1990-91

The material presented here covers Michaelmas Term 1990 (October to December) only.

Course Notes:—

Course 212 (Topology), Academic Year 1990—91, Section 1: Metric Spaces
This section introduces and develops the theory of metric spaces
Course 212 (Topology), Academic Year 1990—91, Section 2: Cauchy Sequences and Completeness
This section develops the theory of complete metric spaces
Course 212 (Topology), Academic Year 1990—91, Section 3: Topological Spaces
This section introduces and develops the theory of topological spaces

Examinations:—

Course 212, Supplemental Examination 1991

## Course 212 (Topology) in the Academic Year 1989-90

The material presented here covers Hilary and Trinity Terms 1990 (January to May). Cross-references to Michaelmas Term material are consistent with course notes for Course 212 in Michaelmas Term 1990.

Course Notes:—

Course 212 (Topology), Academic Year 1989—90, Section 4: Compact Topological Spaces
This section proves various results concerning the topological notion of compactness.
Course 212 (Topology), Academic Year 1989—90, Section 5: Connected Topological Spaces
This section proves various results concerning the topological notions of connectedness and path-connectedness.
Course 212 (Topology), Academic Year 1989—90, Section 6: Smooth Manifolds and Partitions of Unity
This section introduces the concept of a smooth manifold and applies theorems concerning compactness in order to prove results concerning the existence of partitions of unity subordinate to open covers on smooth manifolds.
Course 212 (Topology), Academic Year 1989—90, Section 7: Homotopy and the Fundamental Group
This section introduces homotopies that represent continuous deformations of continuous maps, and introduce the definition of the fundamental group of a topological space.
Course 212 (Topology), Academic Year 1989—90, Section 8: Covering Maps
This section describes the theory of covering maps
Course 212 (Topology), Academic Year 1989—90, Section 9: Winding Numbers
This section develops the theory of winding numbers of closed curves in the complex plane

Examinations:—

Course 212, Scholarship Examination 1990

Dr. David R. Wilkins
School of Mathematics, Trinity College, Dublin 2, Ireland
dwilkins@maths.tcd.ie