620-311 METRIC SPACES

Subject Description :

This subject extends ideas and results about limits and continuity
from Euclidean spaces to very general situations, for example
spaces of functions and manifolds. The material is important throughout analysis, geometry and topology, and has applications
to numerical mathematics, differential and integral equations,
optimisation, physics, logic, computing and algebra.

We begin with a study of Metric Spaces where limits and continuity are defined in terms of a distance function. The need to find limits in a given situation leads to the notions of completeness and compactness. Applications include general results on the solution of
non-linear equations and differential equations.

We will find that many of the arguments do not really need the notion of distance. This leads to the idea of a Topological Space where continuity is defined in terms of "neighbourhoods'' or"open sets''. In this general abstract setting we study the ideas of compactness (which guarantees existence of maxima and minima) and connectedness (which is related to the intermediate value theorem). Finally, we try to understand abstract topological spaces by
studying the existence of continuous functions from these spaces to the real line.

definition, examples, open and closed sets, continuity,
compactness, connectedness, product spaces, Hausdorff spaces, normal spaces, Urysohn's theorem, Tietze's extension theorem.