Synopsis for Topology

Overview

The ideas, concepts and constructions in general topology arose from extending the notions of continuity and convergence on the real line to more general spaces. The first class of general spaces to be studied in this way were metric spaces, a class of spaces which includes many of the spaces used in analysis and geometry. Metric spaces have a distance function which allows the use of geometric intuition and gives them a concrete feel. They allow us to introduce much of the vocabulary used later and to understand the formulation of continuity which motivates the axioms in the definition of an abstract topological space.

The axiomatic formulation of a topology leads to topological proofs of simplicity and clarity often improving on those given for metric spaces using the metric and sequences. There are many examples of topological spaces which do not admit metrics and it is an indication of the naturality of the axioms that the theory has found so many applications in other branches of mathematics and spheres in which mathematical language is used.

Learning Outcomes

The outcome of the course is that a student should understand and appreciate the central results of general topology and metric spaces, sufficient for the main applications in geometry, number theory, analysis and mathematical physics, for example.

Synopsis

Metric spaces. Examples to include metrics derived from a norm on a real vector space, particularly , , norms on , the norm on the bounded real-valued functions on a set, and on the bounded continuous real-valued functions on a metric space. Continuous functions definition). Uniformly continuous functions; examples to include Lipschitz functions and contractions. Open balls, open sets, accumulation points of a set. Completeness (but not completion). Contraction Mapping Theorem. Completeness of the space of bounded real-valued functions on a set, equipped with the norm, and the completeness of the space of bounded continuous real-valued functions on a metric space, equipped with the metric. [3 lectures].

Axiomatic definition of an abstract topological space in terms of open sets. Continuous functions, homeomorphisms. Closed sets. Accumulation points of sets. Closure of a set together with its accumulation points). Interior of a set. Continuity if . Examples to include metric spaces (definition of topological equivalence of metric spaces), discrete and indiscrete topologies, subspace topology, cofinite topology, quotient topology. Base of a topology. Product topology on a product of two spaces and continuity of projections. Hausdorff topology. [5 lectures]

Connected spaces: closure of a connected space is connected, union of connected sets is connected if there is a non-empty intersection, continuous image of a connected space is connected. Path-connectedness implies connectedness. Connected open subset of a normed vector space is path-connected. [2 lectures]

Compact sets, closed subset of a compact set is compact, compact subset of a Hausdorff space is closed. Heine-Borel Theorem in . Product of two compact spaces is compact. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. Equivalence of sequential compactness and abstract compactness in metric spaces. [4 lectures]

Further discussion of quotient spaces explaining some simple classical geometric spaces such as the torus and Klein bottle. [2 lectures]

1. W. A. Sutherland, Introduction to Metric and Topological Spaces (Oxford University Press, 1975). Chapters 2-6, 8, 9.1-9.4.
   (New edition to appear shortly.)
2. J. R. Munkres, Topology, A First Course (Prentice Hall, 1974), chapters 2, 3, 7.

Further Reading

1. B. Mendelson, Introduction to Topology (Allyn and Bacon, 1975). (cheap paperback edition available).
2. G. Buskes, A. Van Rooij, Topological Spaces (Springer, 1997).
3. N. Bourbaki, General Topology (Springer, 1998).
4. J. Dugundji, Topology (Allyn and Bacon, 1966), chapters 3, 4, 5, 6, 7, 9, 11. [Although out of print, available in some libraries.]