A radical approach to algebra by Mary W Gray

By Mary W Gray

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Thus GnH = 0. Similarly, Gntl = 0. Thus G and H are not contiguous. 15 Theorem Let A, B, G and H be sets in a metric space %. If AczG and B e / / , then ^4 and B contiguous => G and // contiguous. Open and closed sets (II) Proof Suppose that AnB^0. 37 Then GnH=>AnB^0. e. non-overlapping) open sets G and H such that A a G and BaH. Proof (i) Suppose that G and H are disjoint open sets for which AaG and B<^H. 15. (ii) Suppose that A and B are separated. Define the sets G and H by G = {x:d(x, A)

Ii) If d(x, E) = 0, then either xeEaE or else xeCE. In the latter case d(x, CE) = d(x9 x) = 0 and so xedEcE. Thus d(x, £) = 0 => xeE. 6 33 Corollary Let S and T be sets in a metric space X. Then Proof Let xeS. 5, d(x, S) = 0. But S c T implies , S)^d{\, T). Hence d(x, T) = 0. Thus x e T . 7f Theorem Let £ be a set in a metric space £. Then £ is the smallest closed set containing E. 10. We therefore assume that dfe, £) = 0 and seek to deduce that £e£. 22, given any £>0, we ean find an xe£ such that d(£, x)

17 Exercise (1) Decide which of the following pairs of sets in (R2 are contiguous and which are separated. (i) A = {(x9 y): x>0}; B = {(x, y): x < 0 } (ii) A = {(x,y):x>0};B = {(x9y):y = 0andx£0} (hi) A = {(x, y): x2 +y2<1}; B = {(x, y): x2 + y2>0} (iv) 4 = {(x, y): x>0 and y ^ O } ; B = {(x, y): x^O and y<0} (v) X = { ( x , ) ; ) : x ^ 0 a n d ) ; ^ 0 } ; B = {(x,>;):x<0and);<0}. (2) Prove that two closed sets A and 5 in a metric space Z are contiguous if and only if they overlap. (3) Suppose that A and B are sets in a metric space % such that AKJB= %.

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