30 1. Various Ways of Representing Surfaces and Examples

No such uniqueness or classification result is possible with metric

spaces and topological spaces in general; these definitions are exam-

ples of the second category of mathematical objects, and are gener-

alities rather than specifics. In and of themselves, they are far too

general to allow any sort of complete classification or universal un-

derstanding, but they have enough properties to allow us to eliminate

much of the tedious case by case analysis, which would otherwise be

necessary when proving facts about the objects in which we are really

interested. The general notion of a group, or of a Banach space, also

falls into this category of generalities.

Before moving on, there are three definitions of which we ought

to remind ourselves. First, recall that a metric space is complete if

every Cauchy sequence converges. This is not a purely topological

property, since we need a metric in order to define Cauchy sequences;

to illustrate this fact, notice that the open interval (0, 1) and the real

line R are homeomorphic, but that the former is not complete, while

the latter is.

Secondly, we say that a metric space (or subset thereof) is com-

pact if every sequence has a convergent subsequence. In the context of

general topological spaces, this property is known as sequential com-

pactness, and the definition of compactness is given as the require-

ment that every open cover have a finite subcover; for our purposes,

since we will be dealing with metric spaces, the two definitions are

equivalent. There is also a notion of precompactness, which requires

every sequence to have a Cauchy subsequence.

The knowledge that X is compact allows us to draw a number

of conclusions; the most commonly used one is that every continuous

function f : X → R is bounded, and in fact achieves its maximum

and minimum. In particular, the product space X × X is compact,

and so the distance function is bounded.

Finally, we say that X is connected if it cannot be written as the

union of non-empty disjoint open sets; that is, if X = A ∪ B, with A

and B open and A ∩ B = ∅, implies either A = X or B = X. There is

also a notion of path connectedness, which requires for any two points

x, y ∈ X the existence of a continuous function f : [0, 1] → X such

that f(0) = x and f(1) = y. As is the case with the two forms of