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By Eells J. (ed.)

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Is E dense in [0, 1 ] ? Is E compact ? Is E perfect ? Is there a nonempty perfect set i n R 1 which contains no rational number ? (a) If A and B are d i sj o i nt closed sets i n some metric space X, prove that they are separa ted . (b) Prove the same for disjoint open sets. (c) Fix p e X, S > 0, define A to be the set of all q e X for which d(p, q) < S, define B similarly, with > in place of < . Prove that A and B are separated . (d) Prove that every connected metric space with at least two points is uncount­ able.

If n is so large that 2 < I such an n, for otherwise 2n < b/r for all positive integers n, which is absurd since R is archimedean), then (c) implies that In c G« , which con­ tradicts (b). This completes the proof. - The equivalence of (a) and (b) in the next theorem is known as the Heine­ Bore! theorem . 41 Theorem If a set E in R k has one of the following three properties, then it has the other two: (a) (b) (c) E is closed and bounded. E is compact. Every infinite subset of E has a limit point in E.

Thus q e{3, and (II) holds. Put t =p + (r/2). Then t > p, and - t - (r/2) =-p- r ¢ a, so that t e{3. Hence {3 satisfies (I II). We have proved that {3e R. If r e a. and s E fJ, then -s ¢ a, hence r < -s, r+ s < 0 . + p c 0*. To prove the opposite i nclusion, pick ve 0* , put w = - vf2. Then w > 0, and there i s an i nteger n such that nw e a but (n + 1 )w¢a.. ) Put p=- ( n + 2)w . +{3. Thus 0* c a + f3. We conclude that a + f3= 0* . This {3 will of cou rse be denoted by - �- Step 5 Havi ng proved that the addition defined in Step 4 sati sfies Axioms (A) of Definition 1 .

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