Graphics Reference
In-Depth Information
CHAPTER 5
Point Set Topology
5.1
Introduction
In this chapter we introduce the basic concepts dealing with metric and topological
spaces and their associated maps. We shall build on the special case of
R
n
as described
in Section 4.2. The reader new to topology can always think in terms of Euclidean
space and its subspaces. That is certainly where currently most of the applications
are. On the other hand, abstract topological spaces are not just abstract nonsense and
it is worthwhile to introduce them even though metric spaces would be adequate for
geometric modeling. The fact is that the metric usually has little to do with anything.
The key concept is that of open sets. It is the open sets that really define a topology
and by studying topological spaces one strips away the unimportant elements and gets
to what is essential for understanding intrinsic topological properties of spaces. It is
inadequate to think of spaces purely in terms of specific imbeddings in
R
n
, even
though, for example, the average person probably only thinks of a circle as an object
sitting in some background like a piece of paper. If we want to study our universe, we
would not think of it as imbedded in another space. The circle and universe have
intrinsic properties that do not depend on any particular imbedding. One needs to see
beyond the imbeddings.
Our intent is to survey only the most important results from what is called
general
or
point set topology
- those that get used a lot in other contexts. There are actually
no really difficult theorems in this chapter. Most follow from the definitions in a rel-
atively straightforward way. Of course, the author realizes that the material here may
well be new to many readers and quite different from what they may have seen before,
so that even easy results may seem hard initially. There are quite a few topics on point
set topology, but the one that this author recommends most highly is Eisenberg's topic
[Eise74]. Most of the explicit references in this chapter will be to this topic, however,
in the case of references for omitted proofs one can find these in many other topics,
such as [Lips65].
Metric spaces are certainly the most important topological spaces and we start
with those in Section 5.2. Section 5.3 defines and discusses general topological spaces.
Section 5.4 describes some important standard operations that create new spaces
