Topology is an important andi nteresting area of mathematics, the study of
which will not only introduce you to new concepts and theorems but also put
into context oldones like continuous functions. However, to say just this is to
understate the significance of topology. It is so fundamental that its influence
is evident in almost every other branch of mathematics. This makes the study
of topology relevant to all who aspire to be mathematicians whether their
first love is (or will be) algebra, analysis, category theory, chaos, continuum
mechanics, dynamics, geometry, industrial mathematics, mathematical biology,
mathematical economics, mathematical finance, mathematical modelling,
mathematical physics, mathematics of communication, number theory,
numerical mathematics, operations research or statistics. Topological notions
like compactness, connectedness and denseness are as basic to mathematicians
of today as sets and functions were to those of last century.
Topology has several different branches — general topology (also known
as point-set topology), algebraic topology, differential topology and topological
algebra — the first, general topology, being the door to the study of the others.
We aim in this book to provide a thorough grounding in general topology.
Anyone who conscientiously studies about the first ten chapters and solves at
least half of the exercises will certainly have such a grounding.
For the reader who has not previously studied an axiomatic branch of
mathematics such as abstract algebra, learning to write proofs will be a hurdle.
To assist you to learn how to write proofs, quite often in the early chapters, we
include an aside which does not form part of the proof but outlines the thought
process which led to the proof. Asides are indicated in the following manner:
In order to arrive at the proof, we went through this thought process,
which might well be calledthe “discovery” or “experiment phase”.
However, the reader will learn that while discovery or experimentation
is often essential, nothing can replace a formal proof.