By A. F. Beardon

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**Sample text**

As an illustration of the ideas involved, we pause to prove (6) and give a particularly attractive application of it. First, we give the proof. Suppose that and that Y X is compact, that f : X -»■ Y is Hausdorff. It is only necessary to prove that sets to open sets. If A is an open subset of and hence, by (3), compact. By (5), f(B) closed. Thus f find that is a continuous bijection Y - f(B) Y - f(B) is is open. 1 Let X = X, then maps open is closed is compact and by (4), it is is a bijection of f(A) f B = X - A X onto Y we is open.

For each rvi\ ]R. (-°°,a) j in J, is also a topology on X. By considering the that contain a given class together with 0 Show that in this topology, C let T^ be a topology on of subsets of smallest topology that contains {x} X, T^ is not closed. X. Show that to be those topologies show that there exists a C. 3 HAUSDORFF SPACES In analytic arguments about topological spaces it is essential to be able to distinguish between distinct points by means of open sets and in general, this is not possible.

Are joined to each other across the final arc m m/2 is odd. if m is even and (m+l)/2 Topologically, we have taken two spheres and made M if m is odd. cuts in each. 2. for the case m = 4. e. a sphere with M-l handles). 2. ,zm ,°° arcs a 1 and join each of these points to a ,a . m 00 yielding two edges of Thus D. D zq is topologically a The two copies D_. of distinct from zq by non-intersecting simple 2(m+l)-gon, D each a. ,a : they are joined to each other 1 m is odd and they are joined to themselves across if is even.