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MAT246: Concepts in Abstract Math
创建日期：2024-08-20 17:30:35
所属分类：数学mathematics
标签： MAT246
MAT246: Concepts in Abstract Math

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MAT246: Concepts in Abstract Math


Open and closed sets

Topology begins by studying the namesake of the open (0 , 1) and closed [0, 1]. For this topic we will exclusively study subsets of the n-dimensional real space Rn.   A crucial subset is the open ball  Ba,r , where a  ∈ Rn  is its center and r > 0 ∈ R is its radius.  Given a subset A ? Rn  we define

Definition 1. A point x is said to be in the interior of A if some ball centered at x is a subset of A.

Definition 2. A point x is said to be a limit point of A if every ball centered at x contains at least one point of A.

Consider the open interval (0, 1).  The points 0 < x < 1 are clearly interior points and the points 0, 1 are clearly limit points. The difference between (0 , 1) and [0, 1] is on one hand that the closed interval contains all its limits points, and on the other hand the point 0, 1 in the closed interval are not interior points. This leads to the following definitions.

Definition 3. A set A is called open if each of its points is an interior point.

Definition 4. A set A is called closed if it contains all of its limit points.

Operations on open/closed sets

Lets see what happens if you apply standard set operations to open/closed sets.

Lemma 1. The union of any number of open sets is open.

Proof.  Given a collection of open sets Ai  we want to prove that i(u)Ai  is also an

open set.  By definition of union a point x belongs to i(u)Ai  if it belongs to one

of the sets Ai.  Since the set Ai  is open some ball around x will be a subset of

Ai , which means it will also be a subset of the union i(u)Ai.

Lemma 2. The intersection of a finite number of open sets is open.

Proof.  Given a collection of open sets A1,..., An  we want to prove that  Ai

is open. By definition a point x in the intersection UAi  belongs to each of the i

sets Ai. For each of those sets there is a ball centered at x that is contained in that set. Since there is a finite number of these balls, the smallest of these ball

will be contained in each of the sets Ai.

We might now want to prove analogous lemmas for closed sets, but instead of doing this directly we will use the following important lemma and de Morgan’s laws.

Lemma 3. A set A is open if and only if its complement Ac  is closed.

Proof.  Given an open set A we want to show that its complement Ac  is closed. Consider a limit point x of the set Ac , if it belongs to A, then there would be a ball Br  centered at x that is a subset of A, in particular this ball would not contain any points of Ac , so it would not be a limit point of Ac.  This shows that every limits point of Ac  must belong to Ac , so Ac  is closed.

Given a set A such that its complement Ac  is closed we want to prove that A is open.  Let x be an arbitrary point in A. We want to find a ball Br  centered at x such that Br  is a subset of A.  If for all r  > 0 we have that Br  is not a subset of A this would mean that for all r > 0 the ball Br  centered atx contains a point in Ac  making x alimit point of Ac , but we assumed that Ac  was closed, so it must contain all its limit points, in particular no point x in A is a limit point of Ac.  This implies that there must exist an r > 0 such that the ball Br centered at x is a subset of A.

This lemma can be restated in the following way.

Lemma 4. The  complement  of an  open set is closed, the complement of a closed set is open.

We can now prove the lemmas.

Lemma 5. An arbitrary intersection of closed sets is closed.

Lemma 6. A finite union of closed sets is closed.

By using de Morgan’s laws and the lemmas we prove previously for open sets.

Proof.  Given a collection of closed sets Ai.  By previous lemma their comple-ments Ai(c) are open. To prove that Ai  is closed we can instead prove that its complement ( Ai )c  which according to de Morgan’s law is equal to i(U)Ai(c) , which is open since it is union of open sets.  A similar argument works for a finite union of closed sets.

Exercise 1. What about infinite unions of closed sets or infinite intersections of open sets?

Basics of Rn

The points in Rn  can be viewed as vectors and you now from linear algebra that vectors have a norm.  The norm of a vector ?x  = (x1,..., xn ) is the same as the distance from the origin to its endpoint x. This norm is denoted as ||x|| and can be expressed as √x1(2) + ... + xn(2) . The crucial property we will need to

prove basic facts about subsets of Rn  is the triangle inequality.

Lemma 7. For any two points x,y  ∈ Rn  we have  Ⅱx + yⅡ ≤ ⅡxⅡ + ⅡyⅡ .

Convergence in Rn

Lets rewrite the definition of a sequence (xn ) converge to a point L  in a way that works for both R and Rn.

Definition 5. A sequence of points (xn ) converges to the points L ifevery ball centered at L contains all but finitely many points of the sequence.

We can also write this in a more familiar fashion using the norm on Rn.

Definition 6. A sequence of points  (xn ) converges to a point L  if for every ε > 0 there exists an N such that for all n > N we have Ⅱxn ? LⅡ < ε .

Exercise 2. Check that these two definitions are equivalent.

It is important to compare the notion of limit to that of a limit point.

Exercise 3. Show that if a sequence (xn ) converges to a point L, then L  is a limit point of the set (xn ).

The converse is not quite true,  even  if we  make  sure the limit point in question is not an element of the sequence.

Exercise 4. Construct  a  sequence which  has two limits points that do not belong to it.

A sequence can have many limits points, but it can only have one limit.

Lemma 8. A sequence of points (xn ) can only have one limit.

Proof.  Suppose a sequence of points (xn ) converges to a point L.  For any other point L′ we can choose small balls one centered at L  the  other  centered at L′ such that these balls do not intersect.  By definition of convergence all but finitely many element of the sequence will belong to the ball centered at L, which implies that only finitely many points can belong to the ball centered at L′ . This means that the sequence does not converge to L′ .

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