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.
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