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. Whole of N is its boundary, Its complement is the set of its exterior points (In the metric space R). Proposition A set O in a metric space is open if and only if each of its points are interior points. 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. Let be a metric space. Definition 1.15. Definition 1.14. The purpose of this chapter is to introduce metric spaces and give some definitions and examples. Defn Suppose (X,d) is a metric space and A is a subset of X. It depends on the topology we adopt. First, if pis a point in a metric space Xand r2 (0;1), the set (A.2) Br(p) = fx2 X: d(x;p) 0. A metric space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in X. 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 . Theorem In a any metric space arbitrary intersections and finite unions of closed sets are closed. Suppose that A⊆ 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! Set Q of all rationals: No interior points. This distance function :×→ℝ must satisfy the following properties: Proof Exercise. Let S be a closed subspace of a complete metric space X. Definition Let E be a subset of a metric space X. Definition 3. We do not develop their theory in detail, and we leave the verifications and proofs as an exercise. In most cases, the proofs 1. 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= ? A point x is called an isolated point of A if x belongs to A but is not a limit point of A. Example: Any bounded subset of 1. Proposition A set C in a metric space is closed if and only if it contains all its limit points. 2. Theorem 9.6 (Metric space is a topological space) Let (X,d)be a metric space. 1.5 Limit Points and Closure As usual, let (X,d) be a metric space. A metric space X is compact if every open cover of X has a finite subcover. Proof. 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. Theorem 4. A closed subset of a complete metric space is a complete sub-space. Interior, Closure, Exterior and Boundary Let (X;d) be a metric space and A ˆX. 10.3 Examples. 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 set Uˆ Xis called open if it contains a neighborhood of each of its (0,1] is not sequentially compact … The closure of A, denoted by A¯, is the union of Aand the set of limit points of A, A¯ = … A subset is called -net if A metric space is called totally bounded if finite -net. 1. A metric space (X,d) is said to be complete if every Cauchy sequence in X converges (to a point in X). Set N of all natural numbers: No interior point. 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. 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