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 the core 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:
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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:
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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 between modern analysis and topology;
discuss the consequences of Urysohn’s lemma;
use examples to explain the significance of Tychonoff’s theorem.
Syllabus:
Introduction [10 hours]:
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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]:
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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 and Banach spaces.
Quotient topology [4 hours]:
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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]:
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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]:
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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]:
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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]:
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Urysohn’s lemma, its consequences and generalization: Tietz’s extension lemma;
paracompact spaces, examples, Stone’s theorem
Tychonoff’s theorem [3 hours]:
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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
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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
http://ourvle.mona.uwi.edu/
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
2. Crossley M. D., Essential Topology, Springer Verlag, 2005.
3. Jaenich K., Topology, Springer Verlag, 2000.
4. 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. http://at.yorku.ca/topology/educ.htm - This website presents a well-developed, new and
growing collection of notes for students learning topology.
2. http://www.geom.uiuc.edu/zoo/ - 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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