every subset of discrete metric space is bounded

Let x∈ A and consider the open ball B(x,1). Consider a metric space (X,d) whose metric d is discrete. On the other hand, if ρ is the discrete metric on an infinite set X, then X is bounded but not totally bounded… 39.4. Every non-empty discrete space is second category. Proposition 2.3 Every totally bounded metric space (and in particular every compact met-ric space) is separable. Show that K is not compact in (X, d). 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. Let and be two metric spaces. 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! The set A is either finite or By Theorem 39.5, Xis open. (5) We have shown that every compact set in a metric space (X, d) is closed and bounded in (X, d). 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. Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain concepts. 3. Prove that every subset of Mis open. Show that every subset A⊂ X is open in X. 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). Let . ... Every function from a discrete metric space is continuous at every point. 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). Each compact metric space is complete, but the converse is false; the simplest example is an infinite discrete space with the trivial metric. 5. Show that the real line is a metric space. Metric Spaces Page 4 . Since d is discrete, this open ball is equal to {x}, so it is contained entirely within A. 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. Now let A = S∞ n=1 An. Start studying Analysis Midterm I. Every discrete space with at least two points is totally disconnected. Likewise, the empty subset ;in any metric space has interior and closure equal to the subset ;. for any metric space X we have int(X) = X and X = X. (iv) By Corol-lary 38.7, X0is closed. Let X be a subset of M. Since M is nite, the complement X0is nite. The moral is that one has to always keep in mind what ambient metric space one is working in when forming interiors and closures! 39.5. Solution. A metric space (X,d) is a set X with a metric d defined on X. Every discrete space is first-countable, and a discrete space is second-countable if and only if it is countable. 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. Let K be an infinite subset of X. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Any two discrete spaces with the same cardinality are homeomorphic. 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. Proof. On the other hand, if we take the real numbers with the discrete metric, then we obtain a bounded metric space. 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. (4) Let (X, d) be a metric space, where d is the discrete metric. Definition. Hence every subset of Mis open. Let Mbe a metric space such that Mis a nite set. 1. These are easy consequences of the de nitions (check!). Every discrete metric space is bounded.

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