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.
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