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