20. The Metric TopologyDefn: Suppose d : X × X → R. Then d is ametric on S if d satisfies the following conditions.1.) d(x, y) ≥ 0 for all (x, y) ∈ X × X;d(x, y) = 0 if and only if x = y.2.) d(x, y) = d(y, x) for all (x, y) ∈ X × X.3.) d(x, z) ≤ d(x, y) + d(y, z)∀x, y, z ∈ X.Example 1 (the euclidean metric on Rn):d1(x, y) =qΣni−1(xi− yi)2where x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn).Example 2 (the square metric on Rn):ρ(x, y) = max{1≤i≤n}|xi− yi|where x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn).Example 3 (the discrete metric on X):d3(x, y) =0 x = y1 x 6= y.19Example 4: Let C = set of all continuous real-valued functions on [0, 1].d4(f, g) = max{|f(x) − g(x)| | x ∈ [0, 1]}.Defn: Bd(p, r) = {x ∈ X | d(p, x) < r}Defn: If d is a metric, then{Bd(p, r) | , p ∈ X, r > 0}is a basis for the metric topology on X inducedby d .Lemma: U is open in the metric topology on Xinduced by d if f or every y ∈ U , there exists anr > 0 such that Bd(y, r) ⊂ U.Defn: If X is a topological space, X is s aid to bemetrizable if there exists a metric d on X whichinduces the topology on X. A metric space is ametrizable space X together with a specific metricd that gives the topology on X.20Defn: Let X be a metric sp ace with me tric d.A subset A of X is bounded if there exists anumber M such that d(a1, a2) ≤ M for everya1, a2∈ A. If A is bounded and nonempty, thediameter of A =diam A = sup{d(a1, a2) | a1, a2∈ A}.Note that boundedness is not a topological prop-erty.Thm 20.1: Let X be a metric space with metricd. Defined : X × X → R byd(x, y) = min{d(x, y), 1}.Thend is a metric that induces the same topologyas d.Defn: The metricd is called the standardbounded metric corresponding to d.21Lemma 20.2: Let d and d′be two metrics on X;let T and T′be the topologies they induce, re-spectively. Then T′is finer tha t T if and onlyif for each x ∈ X and each ǫ > 0, there exists aδ > 0 such that Bd′(x, δ) ⊂ Bd(x, ǫ).Corollary T′is finer that T if there exists a k > 0such that for all x, y ∈ X:Thm 20 .3: The topologies on Rninduced by theeuclidean metric d and the square metric ρ arethe same as the p rodu ct topology on
View Full Document
Unlocking...