Metric Spaces

Definition 1.1 (Cardinal number and equivalence realtion). If there exists a 1 — 1 mapping of A onto B, we say that A and B can be put in 1?— 1 correspondence, or that A and B have the same cardinal number, or, briefly, that A and B are equivalent, and we write A ~ B. This relation clearly has the following properties: It is?reflexive: A ~ A. It is?symmetric: If A ~ B, then B ~ A. It is?transitive: If A ~ B and B ~ C, then A ~ C. Any relation with these three properties is called an equivalence relation.

Definition 1.2?(Countability and Finiteness). For any positive integer n, let J(n) be the set whose elements are the integers 1, 2, ..., n; let J be the set consisting of all positive integers. For any set A, we say:

(a) A is?finite if A ~ J(n) for some n (the empty set is also considered to be finite);

(b) A is?infinite if A is not finite;

(c) A is?countable?if A ~ J;

(d) A is?uncountable?if A is neither finite nor countable;

(e) A is?at most countable if A is finite or countable.

Theorem 1.1. Every infinite subset of a countable set A is countable.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.?Let A be the set of all sequences whose elements are the digits 0 and 1 (i.e., the elements of A are sequences like 1, 0, 0, 1, 0, 1, 1, 1, ...). This set A is uncountable.?

Recall?(Completeness of?R).

(a) Cauchy's criterion:

(b) Compactness:

(c) Any bounded sequence in?R?has convergent subsequence.

(d) Nested interval:

(e) Least upper bound theorem.

Definition 2.1 (Metric space).

Example 1.

Proof.

Example 2.

Example 3.

Definition 3.1?(Convergence and divergence in metric spaces with infinite tuples).

Note that the "=" here is the equivalence of points, not the equivalence of numbers.

Theorem 3.1.

Proof.

Remark 3.1.?Theorem 3.1 does not hold for all metric spaces. As a counterexample:

Note that in the previous counterexample, the?theorem is true from left to right, but wrong from right to left.

Recall (Open ball, closed ball and closure).

Definition 4.1 (Limit point and closed set).

Theorem 4.1.?A finite union of closed subsets (of a metric space) is closed.

Note that it suffies to show that the union of two closed sets is closed.

Counterexample for infinite unions:

Theorem 4.2. An intersection of closed subsets (of a metric space) is closed.

Note that this theorem does not require a finite intersection.

Definition 4.2?(Open set).

Theorem 4.3.

Proof.

Theorem 4.4.

Proof.

Corollary 4.1.?A?finite?intersection of open?subsets (of a metric space) is open.

Note that this corollary follows directly from Theorem 4.1 and De Morgan's Law.

Counterexample for infinite intersections:

Corollary 4.2.?A?union?of open?subsets (of a metric space) is open.

Note that this corollary does not require a finite union, and it?follows directly from Theorem 4.2?and De Morgan's Law.

Definition 4.3 (Topology).?If X is a set and?J?is a collection of subsets of X satisfying:

(i) X,?? ∈?J;

(ii) The union of a subcollection of?J?is a member of J;

(iii) The intersection of a finite subcollection of?J is a member of?J.

Then,?J is called a?topology for X.

Definition 5.1?(Continuous functions in metric spaces).?Let (M1, d1) and?(M2, d2) be metric spaces, let a?∈?M1 and let f be a function from?M1 into?M2. We say that f is continuous at a if for every ε > 0, there exists δ > 0, s.t. if?d1(x,a) < δ, then?d2(f(x),f(a)) < ε.? We say f is?continuous on?M1?if f is continuous at every point of?M1.

Example.

Theorem 5.1.

Proof.

Corollary 5.1. Let f and g be continuous real-valued?functions on a metric space M. Then,

(1) | f | is continuous on M;

(2) f + cg is continuous on M,?? c?∈?R;

(3) f · g is continuous on M;

(4) f / g is continuous on M if g(x) ≠ 0, ??x?∈ M.

Previously, also see?Honors Analysis I (Personal Notes) Week 1-2.

作者：Ark_Donitz

https://www.bilibili.com/read/cv9749839

出處： bilibili

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