Proof Exercise. The closure of A, denoted by A¯, is the union of Aand the set of limit points of A, A¯ = … FACTS A point is interior if and only if it has an open ball that is a subset of the set x 2intA , 9">0;B "(x) ˆA A point is in the closure if and only if any open ball around it intersects the set x 2A , 8">0;B "(x) \A 6= ? Theorem 4. This distance function :×→ℝ must satisfy the following properties: Set N of all natural numbers: No interior point. Suppose that A⊆ X. In the standard topology or $\mathbb{R}$ it is $\operatorname{int}\mathbb{Q}=\varnothing$ because there is no basic open set (open interval of the form $(a,b)$) inside $\mathbb{Q}$ and $\mathrm{cl}\mathbb{Q}=\mathbb{R}$ because every real number can be written as the limit of a sequence of rational numbers. A point x is called an isolated point of A if x belongs to A but is not a limit point of A. The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. Example: Any bounded subset of 1. The family Cof subsets of (X,d)deﬁned in Deﬁnition 9.10 above satisﬁes the following four properties, and hence (X,C)is a topological space. Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. Definition Let E be a subset of a metric space X. A metric space X is compact if every open cover of X has a ﬁnite subcover. 2. Definition. Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. A metric space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in X. 1. 1. A set Uˆ Xis called open if it contains a neighborhood of each of its Set Q of all rationals: No interior points. The point x o ∈ Xis a limit point of Aif for every neighborhood U(x o, ) of x o, the set U(x o, ) is an inﬁnite set. A subset is called -net if A metric space is called totally bounded if finite -net. Deﬁnition 1.14. Whole of N is its boundary, Its complement is the set of its exterior points (In the metric space R). Defn Suppose (X,d) is a metric space and A is a subset of X. In most cases, the proofs 10.3 Examples. Theorem In a any metric space arbitrary intersections and finite unions of closed sets are closed. It depends on the topology we adopt. Proposition A set O in a metric space is open if and only if each of its points are interior points. Let S be a closed subspace of a complete metric space X. If a subset of a metric space is not closed, this subset can not be sequentially compact: just consider a sequence converging to a point outside of the subset! We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. Proposition A set C in a metric space is closed if and only if it contains all its limit points. Let be a metric space. A point p is a limit point of the set E if every neighbourhood of p contains a point q ≠ p such that q ∈ E. Theorem Let E be a subset of a metric space … Deﬁnition 1.15. Deﬁnition 3. (0,1] is not sequentially compact … Metric Spaces, Open Balls, and Limit Points DEFINITION: A set , whose elements we shall call points, is said to be a metric space if with any two points and of there is associated a real number ( , ) called the distance from to . 1.5 Limit Points and Closure As usual, let (X,d) be a metric space. Proof. First, if pis a point in a metric space Xand r2 (0;1), the set (A.2) Br(p) = fx2 X: d(x;p)

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