Lecture 4 29

The last of these is generally the most interesting, and is sometimes

useful in the following equivalent form:

d(x, y) ≥ |d(x, z) − d(y, z)|.

Once we have defined a metric on a space X, we immediately

have a topology on X induced by that metric. The ball in X with

centre x and radius r is given by

B(x, r) = { y ∈ X | d(x, y) r }.

Then a set A ⊂ X is said to be open if for every x ∈ A, there exists

r 0 such that B(x, r) ⊂ A, and A is closed if its complement X \ A

is open. We now have two equivalent notions of convergence: in the

metric sense, xn → x if d(xn, x) → 0, while the topological definition

requires that for every open set U containing x, there exist some N

such that for every n N, we have xn ∈ U. It is not hard to see that

these are equivalent.

Similarly for the definition of continuity; we say that a function

f : X → Y is continuous if xn → x implies f(xn) → f(x). The

equivalent definition in more topological language is that continuity

requires

f−1(U)

⊂ X to be open whenever U ⊂ Y is open. We say

that f is a homeomorphism if it is a bijection and if both f and

f−1

are continuous.

Exercise 1.14. Show that the two sets of definitions (metric and

topological) in the previous two paragraphs are equivalent.

Within mathematics, there are two broad categories of concepts

and definitions with which we are concerned. In the first instance, we

seek to fully describe and understand a particular kind of structure.

We make a particular definition or construction, and then seek to

either show that there is only one object (up to some appropriate

notion of isomorphism) which fits our definition, or to give some sort

of classification which exhausts all the possibilities. Examples of this

approach include Euclidean space, which is unique once we specify

dimension, or Jordan normal form, which is unique for a given matrix

up to a permutation of the basis vectors, as well as finite simple

groups, or semisimple Lie algebras, for which we can (eventually)

obtain a complete classification.