Department of Mathematics

Course Title:General Topology

Course Code: MATH6621

Semester:I

No. of credits:4

Prerequisites:

None

Rationale:

Topology is the study of spaces and sets and can be thought of as an extension of geometry. It is an investigation of both the local and the global structure of a space or set. The foundation of General Topology (or Point-Set Topology) is set theory. The motivation behind topology is that some geometric problems do not depend on the exact shape of an object but on the way the object is put together. For example, the square and the circle are geometrically different, but they have many properties in common: they are both one dimensional objects and both separate the plane into two parts. Similarly, a donut and a coffee cup are topologically the same even though they look completely different. Much of the study of topology comes from setting aside our preconceived notions of "shape" involving size, length, flat, straight, or curved, and realizing that a circle and a square are really the same thing.

Topological spaces show up naturally in almost every branch of mathematics. A course in general topology is essential for students enrolled in the Master Programme at the Department of Mathematics since it provides them with fundamental notions such as those of topological space, topological vector space, connectedness, and compactness which are fundamental concepts permeating thecore course MATH6110 (Functional Analysis).

Course description:

The course gives an up-to-date and modern overview of the main concepts in General Topology. Topological properties and several examples of topological spaces arising in several branches of mathematics are studied to show how topology is a unifying theme in different mathematical fields.

Content:

Topological, metric and normed spaces – Continuity – Connectedness – Hausdorff’ spaces - Compactness - Completeness - Topological vector spaces – Quotient spaces – Completion of maps, metric and normed spaces – Homotopy – Countability axioms and their role in mathematics – Urysohn’s lemma – Tietz’s extension lemma – Paracompact spaces and Stone’s theorem – Tychonoff’s theorem and its role in Functional Analysis.

Objectives:

By the end of the course, students will be able to:

  • explain the characteristics of topological, metric and normed spaces;
  • discuss the implications of the cardinality of the continuum;
  • construct new spaces from old, including subspaces, quotients and product spaces;
  • construct continuous functions between topological spaces;
  • test convergence of sequences in different spaces;
  • identify connections betweenmodern analysis and topology;
  • discuss the consequences of Urysohn’s lemma;
  • use examples to explain the significance of Tychonoff’s theorem.

Syllabus:

Introduction [10 hours]:

-Cardinality of N versus R;

-The concept of topological space in terms of open sets, closed sets, neighborhoods, and closures; metric spaces and their topology, metrizable spaces; topological subspaces, induced topology, sums and products of topological spaces, product topology; bases and subbases, refinements, trivial and discrete topology.

-Continuous maps between topological spaces, homeomorphisms and homeomorphic spaces; connected topological spaces, path-connected topological spaces, examples.

-Hausdorff’spaces and the separability axiom, convergent sequences in a topological space and in a Hausdorff’ space; compact topological spaces, examples and main theorems.

Topological vector spaces [4 hours]:

-Topological vector spaces; standard topology and finite dimensional vector spaces, examples; Cauchy sequences and complete metric spaces; pseudo-norm, norm and normed spaces; examples of topological vector spaces: Hilbert andBanach spaces.

Quotient topology [4 hours]:

-Quotient space and quotient topology; topological properties inherited by the quotient space; examples: homogeneous spaces, orbit spaces, gluing topological spaces, Moebius strip, Klein bottle.

Completion [3 hours]:

-Completeness and completion, uniqueness and existence theorem of the completion of a metric space, examples; the completion of a normed space is a Banach space, examples.

Homotopy [4 hours]:

-Homotopy maps, their composition and products, homotopy inverse, homotopy equivalence, contractibility of a topological space to a point, deformation retract, strong deformation retract, examples.

-Categories, the homotopy category, functors, invariants, Euler and Betti numbers; the role of homotopy invariance in algebraic topology and examples of reduction of geometric problems to homotopy problems.

Countability axioms [4 hours]:

-First and second countability axioms, examples; the role of the first countability axiom in the sequential characterization of continuity and in the sequential characterization of compactness, the role of the second countability axiom in developing the concept of a manifold.

Urysohn’s lemma [4 hours]:

-Urysohn’s lemma, its consequences and generalization: Tietz’s extension lemma; paracompact spaces, examples, Stone’s theorem

Tychonoff’s theorem [3 hours]:

-Tychonoff’s theorem, motivation of the theorem by means of examples; its application in Functional Analysis.

Tutorials [12 hours]

Teaching methodology:

The abstract concepts, illustrated with examples, will be presented during the lectures. However, the course is designed in such a way to maximize the extent to which students discover the main concepts by themselves. This can be achieved through class participation in discussions during the lectures and tutorial periods guided by the students themselves and supervised by the lecturer. The advantage is that students feel ownership of the ideas they have worked through themselves. Homework problems will be divided into two types: practice problems, which reinforce the basic concepts and are essentially routine and challenging problems, whose resolution will be fundamentally more involved.

Course material, including class notes and practice problems, will be posted on the webpage

ASSESSMENT:

The course assessment has three components:

  1. Two in-course tests - 40% of overall grade;
  2. Final exam - 60% of overall grade.

The final exam will be three hours in length and consist of four compulsory questions.

Reference material:

Books:

Prescribed

  1. Dixmier J.,General Topology, Springer Verlag, 2010.

Recommended readings

  1. Crossley M. D., Essential Topology, Springer Verlag, 2005.
  2. Jaenich K., Topology, Springer Verlag, 2000.
  3. Bourbaki N., General Topology: Chapters 1 – 4, Springer Verlag, 1989.

These books are pedagogically excellent, comprehensively address all element of the syllabus, and provide useful examples.

Online Resources:

  1. - This website presents a well-developed, new and growing collection of notes for students learning topology.
  2. - The Topological Zoo is an ongoing project which is primarily the work of graduate students from at the University of Minnesota. a resource for mathematicians and educators. It is a visual dictionary of surfaces and other mathematical objects, consisting primarily of movies, still images and interactive pictures.

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