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