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Metric Spaces, Continuity, Limit Points

Metric Spaces, Continuity, Limit Points
Course

Numerical Analysis 1 (APM21M1)

49 Documents
Students shared 49 documents in this course
Academic year: 2020/2021
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Walter Sisulu University

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Lecture 1

1 Review of Topology

1 Metric Spaces

Definition 1. Let X be a set. Define the Cartesian product X × X = {(x, y) : x, y ∈ X}.

Definition 1. Let d : X × X →R be a mapping. The mapping d is a metric on X if the following four conditions hold for all x, y, z ∈ X:

(i) d(x, y) = d(y, x),

(ii) d(x, y) ≥ 0,

(iii) d(x, y) = 0 ⇐⇒ x = y, and

(iv) d(x, z) ≤ d(x, y) + d(y, z).

Given a metric d on X, the pair (X, d) is called a metric space.

Suppose d is a metric on X and that Y ⊆ X. Then there is an automatic metric dY on Y defined by restricting d to the subspace Y × Y ,

dY = d Y| × Y. (1) Together with Y , the metric dY defines the automatic metric space (Y, dY ).

1 Open and Closed Sets

In this section we review some basic definitions and propositions in topology. We review open sets, closed sets, norms, continuity, and closure. Throughout this section, we let (X, d) be a metric space unless otherwise specified. One of the basic notions of topology is that of the open set. To define an open set, we first define the �-neighborhood.

Definition 1. Given a point xo ∈ X, and a real number � > 0, we define

U (xo, �) = {x ∈ X : d(x, xo) < �}. (1)

We call U (xo, �) the �-neighborhood of xo in X.

Given a subset Y ⊆ X, the �-neighborhood of xo in Y is just U (xo, �) ∩ Y.

Definition 1. A subset U of X is open if for every xo ∈ U there exists a real number � > 0 such that U (xo, �) ⊆ U.

⋂

We make some propositions about the union and intersections of open sets. We omit the proofs, which are fairly straightforward. The following Proposition states that arbitrary unions of open sets are open.

Proposition 1. Let {Uα, α ∈ I} be a collection of open sets in X, where I is just a labeling set that can be finite or infinite. Then, the set ⋃ Uα is open. α∈I The following Corollary is an application of the above Proposition.

Corollary 1. If Y ⊂ X and A is open in Y (w.r. dY ), then there exists on open set U in X such that U ∩ Y = A.

Proof. The set A is open in Y. So, for any p ∈ A there exists an �p > 0 such that U (p, �p) ∩ Y ⊆ A. We construct a set U containing A by taking the union of the sets U (p, �p) over all p in A, ⋃

U = U (p, �p). (1) p∈A

For every p ∈ A, we have U (p, �p)∩Y ⊆ A, which shows that U ∩Y ⊆ A. Furthermore, the union is over all p ∈ A, so A ⊆ U , which implies that A ⊆ U ∩ Y. This shows that U ∩ Y = A. To conclude the proof, we see that U is open by the openness of the U (p, �p) and the above theorem.

The following Proposition states that finite intersections of open sets are open.

Proposition 1. Let {Ui, i = 1,... , N } be a finite collection of open sets in X. Then the set i=N Ui is open. i=

Definition 1. Define the complement of A in X to be A c = X − A = {x ∈ X : x /∈ A}.

We use the complement to define closed sets.

Definition 1. The set A is closed in X if Ac is open in X.

1 Metrics on Rn

For most of this course, we will only consider the case X = Rn or X equals certain subsets of Rn called manifolds, which we will define later. There are two interesting metrics on Rn. They are the Euclidean metric and the sup metric, and are defined in terms of the Euclidean norm and the sup norm, respectively.

1 Limit Points and Closure

As usual, let (X, d) be a metric space.

Definition 1. Suppose that A ⊆ X. The point xo ∈ X is a limit point of A if for every �-neighborhood U (xo, �) of xo, the set U (xo, �) is an infinite set.

Definition 1. The closure of A, denoted by A ̄, is the union of A and the set of limit points of A,

A ̄ = A ∪ {xo ∈ X : xo is a limit point of A}. (1)

Now we define the interior, exterior, and the boundary of a set in terms of open sets. In the following, we denote the complement of A by A c = X − A.

Definition 1. The set Int A ≡ (A ̄c)c (1)

is called the interior of A.

It follows that

x ∈ Int A ⇐⇒ ∃� > 0 such that U (x, �) ⊂ A. (1) Note that the interior of A is open. We define the exterior of a set in terms of the interior of the set.

Definition 1. The exterior of A is defined to be Ext A ≡ Int A c.

The boundary of a set is the collection of all points not in the interior or exterior.

Definition 1. The boundary of A is defined to be Bd A ≡ X −((Ext A)∪(Int A)).

Always, we have X = Int A ∪ Ext A ∪ Bd A.

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Metric Spaces, Continuity, Limit Points

Course: Numerical Analysis 1 (APM21M1)

49 Documents
Students shared 49 documents in this course

University: Walter Sisulu University

Was this document helpful?
Lecture 1
1 Review of Topology
1.1 Metric Spaces
Definition 1.1. Let X be a set. Define the Cartesian product X× X = { (x, y) :
x, y ∈ X} .
Definition 1.2. Let d: X× X R be a mapping. The mapping dis a metric on →
X if the following four conditions hold for all x, y, z ∈ X:
(i) d(x, y) = d(y, x),
(ii) d(x, y)≥ 0,
(iii) d(x, y) = 0 ⇐⇒ x= y, and
(iv) d(x, z) ≤ d(x, y) + d(y, z).
Given a metric don X, the pair (X, d) is called a metric space.
Suppose dis a metric on X and that Y ⊆ X. Then there is an automatic metric
dY on Y defined by restricting dto the subspace Y × Y,
dY = d Y × Y. (1.1) |
Together with Y, the metric dY defines the automatic metric space (Y, dY ).
1.2 Open and Closed Sets
In this section we review some basic definitions and propositions in topology. We
review open sets, closed sets, norms, continuity, and closure. Throughout this section,
we let (X, d) be a metric space unless otherwise specified.
One of the basic notions of topology is that of the open set. To define an open
set, we first define the �-neighborhood.
Definition 1.3. Given a point xo ∈ X, and a real number � > 0, we define
U(xo, �) = { x∈ X: d(x, xo) < �} . (1.2)
We call U(xo, �) the �-neighborhood of xo in X.
Given a subset Y ⊆ X, the �-neighborhood of xo in Y is just U(xo, �)∩ Y.
Definition 1.4. A subset U of X is open if for every xo U there exists a real ∈
number � > 0 such that U(xo, �)⊆ U.
1

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