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530 Hwk 3-4 - Math 530 “wk 3 From Mun kres page 51/5 page...

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Unformatted text preview: Math 530 “wk 3 From Mun kres : page 51/5 page tit/1.3.4 Math 530 lip-ls; 4 From Munkries: page 125lla page l2t'iv129r‘9-l l 1. Ex.) UI Let <X. S 3’ he a. linearly ordered set with at least two elements. and let no and ho be the smallest and largest elements of X it‘ they exist. We define a subset U to be open in the order topology T for X provided (ll it‘ x E U and x at Eartha}. then 3 an order interval (alt) with x E (ab) ,: U; (I) it‘ an E U. then there exists h E K with Ianb) C U; (Pal it'bo E U. then there exists a E X with tabnl C U. (a) Prove that r isindeedatopology. (b) Prove that V x. the "order rays" (31.00) and (—le are open sets. Consider [0.11 K [0.11 with the lexicographic or dictionary order. Which ot‘the following sets are open in the order topology ( see the last. Exercise)? ta) [0.1! X ll‘; ([3) l0.” X {l}: (c) {tx.yl : Xi‘y}. Show thatasubset A ot‘a metric space <X.d> ishounded provided A is contained in some ball. let <X..d> bca metric space. Let xn E X and or. > t) be fixed. Prove that {x E X : (Kama) > tit} is an open set. in the plane with the square metric. prove-that {(x.yl : x at y} is open. Is it open with respect to the usual metri c? in the. plane. define the distance {3 from t X! .y H to tap/3) to be the usual Euclidean distance provided the two points lie on a common ray emanating from (0.0). and the sum . . . . . . ‘7 ‘7 7 1‘ ot their distances to the origin othcrvvise: ”\liq + y’,’ + \[xg t ya (a) Assuming the Euclidean metric isa metric. prove that p isa metric.cal|ed the French railroad metric. Why might this name be reasonable? (b) Sketch the balls BHtt3.4).Jl. Bptt0.0}.1). B;,trtl3.4).7l, and Bptl.3.—'l).12) (c) Does this metric give a coarser or finer topology than the Euclidean metric? ...
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