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EX. three The kinfolk {Bd(y, r) l (y 6 Y) /\ (r is rational)} is usually a foundation for 701), as is speedy from III, 2. 2. The communicate query arises: Given a topological house (Y, . 7), is there a metric d in Y such that ,7 = . 701)? the answer's no commonly. for instance, you could see that no metric can set off the topology in Sierpinski area. 23 Definition A topological area (Y, . 7) is termed a metric (or metrizable) area if its topology is that precipitated via a metric in Y. A metric for an area Y is one who induces its topology. With this terminology, the Euclidean area E n is a metric area, and do is a metric for this house. In metric areas, topological techniques may be phrased within the s, three phrases of classical research. for instance, 2-4 LetXhave topology 9(5)? ) and Yhave topology Anf. X > Y is continuing if VxVe > 03 3(8, x) > zero: d(§, x) < three => that's, iff[Bd(x, 8)] C Bl,[f(x), a]. < e; 184 Chap. IX Ex. four Metric areas it's fast from the triangle inequality that the formulation I d(x,y) d(x ,y ) l S d(x, ninety six ) + 11(3), y ) is legitimate for any metric d. utilizing this, the reader can simply end up that if (Y, 7(d)) is any metric area and if the cartesian product topology is utilized in Y X Y, then the map d : Y x Y > E1 is constant (and, actually, an identi cation. ) three. similar Metrics during this part we supply acriterion for making a choice on upfront even if diversified metrics in a collection Y will result in a similar topology. three. l De nition metrics d, p, in a collection Y are referred to as an identical, d~ p,if§'(d) = 9(p). this can be essentially an equivalence relation within the set of all metrics on Y. three. 2 Theorem enable p, d, be metrics on Y. an important and suf cient that p ~ d is that for every a e Y and e > zero, the subsequent stipulations carry: (1). El 31 = 31(a, a): p(a, y) < 31 => d(a, y) < a. (2). El 32 = 82(a, a): d(a, y) < 32 => p(a, y) < a. evidence: this is often easily III, three. four, or equivalently, metric statements (cf. 2. four) that 1: (Y, > (Y, is a homeomorphism. Ex. 1 within the set E n, all of the metrics (11,, p > 1 in I, Ex. three, are comparable to the do in I, Ex. 2, and as a result all metrize the Euclidean topology: this follows from 2, Ex. 2and the observations that, for every p > 1, and three. three (a). dp zero there's a metric pM ~ d such that pM(x, y) S M for all (x,y). Equivalently, every one metric area is homeomorphic to a bounded metric area. evidence: Given M, de ne pM(x, y) = min[M, d(x, y)]; it really is trivial to make sure that pM is certainly a metric for Y and, utilizing three. 2, that d ~ pM. four. four. l Continuity of the gap De nition In a metric house Y, with metric d, (1). the space of some degree yo to a nonempty set A is d(yo, A) = inf{d(yo, sixty one) l a E A}- Sec. five homes of Metric Topologies 185 (2). the gap among nonempty units A and B is d(A, B) = inf {d(a, b) I a e A, b e B} = inf {d(a, B) I a e A}. (3). The diameter of a nonempty set A is 3(A) = sup {d(x, y) I x eA, y e A}. EX. 1 3(Bd(a, r)) S 2r, due to the fact that if x, y e Bd(a, r), then d(x,y) S d(x, a) + d(a,y) < 2r; as 2, EX.

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