Show that the real line is a metric space. Let Mbe a metric space such that Mis a nite set. Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain concepts. any nonempty set X, the discrete metric is a bounded metric (it is bounded by 1), and hence every subset of a discrete metric space is bounded. Likewise, the empty subset ;in any metric space has interior and closure equal to the subset ;. Hence every subset of Mis open. Let and be two metric spaces. The set A is either finite or Proof. Show that every subset A⊂ X is open in X. For a metric space let us consider the space of all nonempty closed bounded subset of with the following metric: Check that it is well-defined and a metric! Every non-empty discrete space is second category. Learn vocabulary, terms, and more with flashcards, games, and other study tools. In particular, R is a bounded set w.r.to the discrete metric on R. (iii) {(x,y) ∈ R2: x+ y≤ 1} is an unbounded subset of (R2,dE). 3. Start studying Analysis Midterm I. Now let A = S∞ n=1 An. Show that K is not compact in (X, d). A subset A of a metric space is called totally bounded if, for every r > 0, A can be covered by finitely many open balls of radius r. For example, a bounded subset of the real line is totally bounded. Consider a metric space (X,d) whose metric d is discrete. Let . Every discrete space with at least two points is totally disconnected. 1. Let K be an infinite subset of X. (iv) (5) We have shown that every compact set in a metric space (X, d) is closed and bounded in (X, d). By Theorem 39.5, Xis open. These are easy consequences of the de nitions (check!). The moral is that one has to always keep in mind what ambient metric space one is working in when forming interiors and closures! 5. Let X be a subset of M. Since M is nite, the complement X0is nite. Metric Spaces Page 4 . Each compact metric space is complete, but the converse is false; the simplest example is an infinite discrete space with the trivial metric. Solution. Definition. We can define many different metrics on the same set, but if the metric on X is clear from the context, we refer to X as a metric space and omit explicit mention of the metric d. Example 7.2. A metric space (X,d) is a set X with a metric d defined on X. We take any set Xand on it the so-called discrete metric for X, de ned by d(x;y) = (1 if x6=y; 0 if x= y: This space (X;d) is called a discrete metric space. Further, a metric space is compact if and only if each real-valued continuous function on it is bounded (and attains its least and greatest values). Since d is discrete, this open ball is equal to {x}, so it is contained entirely within A. Proposition 2.3 Every totally bounded metric space (and in particular every compact met-ric space) is separable. 39.5. On the other hand, if we take the real numbers with the discrete metric, then we obtain a bounded metric space. It is not hard to see that a subset of the real numbers is bounded in the sense of if and only if it is bounded as a subset of the metric space of real numbers with the standard metric. Any two discrete spaces with the same cardinality are homeomorphic. (4) Let (X, d) be a metric space, where d is the discrete metric. On the other hand, if ρ is the discrete metric on an infinite set X, then X is bounded but not totally bounded… for any metric space X we have int(X) = X and X = X. By Corol-lary 38.7, X0is closed. Let x∈ A and consider the open ball B(x,1). ... Every function from a discrete metric space is continuous at every point. Every discrete space is first-countable, and a discrete space is second-countable if and only if it is countable. Every discrete metric space is bounded. If X is totally bounded, then there exists for each n a finite subset An ⊆ X such that, for every x ∈ X, d(x,An) < 1/n. Prove that every subset of Mis open. 39.4.
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