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. 9 At least for the usual definition of dimension we mention an alternate definition in the next section.

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