Lecture 4 31

compactness above, these are not equivalent for arbitrary topological

spaces (or even for arbitrary metric spaces—the usual counterexample

is the union of the graph of sin(1/x) with the vertical axis), but will

be equivalent on the class of spaces with which we are concerned.

b. Homeomorphisms and isometries. In the topological context,

the natural notion of equivalence between two spaces is that of home-

omorphism, which we defined above as a continuous bijection with

continuous inverse. Two topological spaces are homeomorphic if there

exists a homeomorphism between them. Any property common to all

homeomorphic spaces is called a topological invariant; this naturally

includes any property defined in purely topological terms, such as

connectedness, path-connectedness, and compactness.

Some invariants require a little more work; for example, we would

like to believe that dimension is a topological invariant, and this is

in fact true,9 but proving that Rm and Rn are not homeomorphic for

m = n requires non-trivial tools.

A considerable part of this course deals with topological invari-

ants of compact surfaces, and in particular, the task of classifying

such surfaces up to a homeomorphism. We will almost succeed in

solving this problem completely; the only assumption we will have

to make is that the surfaces in question admit one of several natural

additional structures. In fact this assumption turns out to be true for

any surface, but we do not prove this in this course.

The natural equivalence relation in the geometric setting is isom-

etry; a map f : X → Y between metric spaces is isometric if

dY (f(x1), f(x2)) = dX (x1, x2)

for every x1, x2 ∈ X. If in addition f is a bijection, we say that f is

an isometry. We are particularly interested in the set of isometries

from X to itself,

Isom(X, d) = { f : X → X | f is an isometry },

which we can think of as the symmetries of X. In general, the more

symmetric X is, the larger this set.

9At

least for the usual definition of dimension; we mention an alternate definition

in the next section.