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from old ones. If one had to list the two most used properties of topological spaces it
would certainly be compactness and connectedness and we devote Sections 5.5 and
5.6 to those, respectively. The basic problem of topology is to classify spaces up to
homeomorphism and to find invariants that can be used to distinguish homeomor-
phism classes. However, along with deformations of spaces, it is also useful to study
deformations of mappings and we do this in Section 5.7 where we discuss homotopy.
Section 5.8 describes conditions for the existence of certain continuous functions. In
Section 5.9 we take another look at a very important space, P n , and discuss some of
its topological properties.
Finally, point set topology is one of those fields where one encounters a great many
definitions. The reader may start to feel overwhelmed by all these definitions at the
first reading of this chapter. In some sense, the reader can “ignore” them until they
become relevant in the context of specific results. The reader may also run into many
of the terms elsewhere, and so this chapter will serve as a general reference for what
they mean. Certainly, we had to present them here because they represent certain
technical conditions without which theorems would be false. The reader who is learn-
ing about topology for the first time may wonder “what the fuss is all about” because
the conditions might seem like conditions that should obviously hold. They may in
fact hold for all the “nice” spaces we will ever consider. However, definitions by their
nature are abstract and they may have consequences that were unintended. For
example, the definition of a continuous function is one with which “everyone” is happy
and which can be used very effectively, but there are continuous functions that are
nowhere differentiable. Is that what one had in mind when defining a continuous
function? (As an aside, although such functions are undesirable, their existence actu-
ally gives us insight into what continuity really means.) The same thing is true here.
Point set topology is a very large field. The definitions that we shall deal with in this
topic only scratch the surface. The fact is that, like the continuity of functions, the
definition of a topology on a space is abstract and although it captures a basic idea
that was extrapolated from nice subspaces of R n , it allows for a much larger universe
of spaces. This is why we shall have to introduce some additional conditions from
time to time (like requiring the differentiability of a function in addition to its conti-
nuity) to guarantee that we get what we want. Isolating needed conditions for a result
and giving them a name is helpful in understanding what makes a theorem true.
Unfortunately, this topic will often not have time to explain fully the nuances and
reasons for naming special conditions leaving some readers with a feeling of mystery.
The only cure for such feelings is to read one of the more comprehensive topics on
topology listed in the references.
5.2
Metric Spaces
Definition.
A metric on a set X is a function
d: XX R
¥Æ
such that the following holds for all p , q , r
X :
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