By A. F. Beardon

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E. closed and bounded) sets whose union is Let d D. 2 . d(f ,f) -*• O n 00 if and only if n TOPOLOGICAL SPACES (which, by convention, we call are in X is a class T of subsets of open sets) with the properties (1) 0,X (2) any finite intersection of sets in (3) any union of sets in T ; T is also in T Often, we refer to a topological space to some unspecified T. is also in T; T. A topological space is a pair X. as D. A topology on a non-empty set X K . n f (X,T) where T is a topology on X: this is an implicit reference A metric space is a topological space, a set being open if and only if it is a union of open balls: this is the metric topology induced by the metric.

1 that the class of open rectangles is a base for some topology on Z: we call this the product topology on Z. There are natural coordinate maps P 1 : (x,y) of Z onto X and Y h- x It follows that if h- y respectively and these are continuous because (p )_ 1 (A) = A x Y Plf , P 2^« P 2 : (x,y) , , f : W -*■ Z (p2)_ 1 (B) = X * B . is continuous, then so are the compositions Conversely, if these compositions are continuous, then so is f because Uf f ' ^ V A a >

This is the equivalence class containing (z,w) is : t € £, t * O}. maps W onto the space the quotient topology induced by IP q : W of equivalence HP. We complex projective space. 2. 4*2. Again, we can transfer the atlas on this yields IP to 3P and using g as a Riemann surface with atlas [z,w] h* z/w on U1 = {[z,w] : w * 0 } , [z,w] ^ w/z on U 2 = ([z,w] : z * O}. 4 1. 1) and that = P 2. Show that 2 z-w I p(w)| d(z,w) = |p(z) (1+1 2 [2)1/2 (l-t-lwl 2)1/2 is a metric on We call (E^ and that the metric topology is the given topology.