NoteNextra-origin/content/Math4111/Math4111_L7.md

Lecture 7

Review

Let S=\{(x,y,z)\in \mathbb{R}^3:x=1,y=4\}=\{(1,4,z):z\in\mathbb{R}\}

1. How can we describe the set S geometrically in three-dimensional space?

   Just a line

2. Show that S and \mathbb{R} are in one-to-one correspondence.

   We can find a bijective function f:S\to \mathbb{R}

3. Show that for any (a,b)\in\mathbb{Z}^2, the set \{(a,b,z):z\in\mathbb{Z}\} is in one-to-one correspondence with \mathbb{Z}

   Use Theorem 2.13 A is countable, n\in \mathbb{N} \implies A^n=\{(a_{1},...,a_{n}):a_1\in A, a_n\in A\}, is countable.

New materials

Metric spaces

Definition 2.15

Let X be a set. A function d:X\times X\to \mathbb{R} is called a distance function or a metric if it satisfies:

1. Positivity: \forall p,q\in X,p\neq q\implies d(p,q)>0, and \forall p\in X,d(p,p)=0.
2. Symmetry: \forall p,q\in X, d(p,q)=d(q,p).
3. Triangle inequality: \forall p,q,r\in X, d(p,q)\leq d(p,r)+d(r,q)

We say (X,d) is a metric space. If d is understood, X is a metric space.

Examples:

The most important example:

X\subset \mathbb{R}^k(k\geq 1)

d(x,y)=|x-y|

And other examples: function spaces...

An example from graph theory (not needed for this class):

d(p,q) can be defined by the shortest path fro p to q.

Definition 2.17

By the segment (a,b) we mean the set of all real numbers x such that a<x<b.

segment excludes the bound (a,b)

By the interval [a,b] we mean the set of all real numbers x such that a\leq x\leq b

• interval include the bound* [a,b]

Convex: E\subset \mathbb{R}^k is convex if \forall x,y\in E,\{\lambda x+(1-\lambda)y:\lambda\in (0,1)\}\subset E

Open sets

Definition 2.18

Let (X,d) be a metric space.

1. p\in X,r>0. The r-neighborhood of p is B_r(p)=N_r(o)=\{q\in X: d(p,q)<r\} (a ball in metric space)
2. E\subset X, p\in X. We say p is an interior point of E if \exists r>0 such that B_r(p)\subset E. Notation $E^{\circ}=$set of interior points of E
3. E\subset X, we say E is open if E\subset E^{\circ}, i.e. \forall p\in E, \exists r>0 such that B_r(p)\subset E.

Note: is follows from definitions that E^{\circ}\subset E is always true.

Example:

$X=\mathbb{R}^2$(d be the euclidean distance) E=[0,1)\times [0,1).

E^{\circ}=(0,1)\times (0,1)

So E=(0,1)\times (0,1) is a open set.

Theorem 2.19

Let (X,d) be a metric space, \forall p\in X,\forall r>0, B_r(p) is an open set.

every ball is an open set

Proof

Let q\in B_r(p).

Let h=r-d(p,q).

Since q\in B_r(p),h>0. We claim that B_h(q). Then d(q,s)<h, so d(p,s)\leq d(p,q)+d(q,s)<d(p,q)+h=r. (using triangle inequality) So S\in B_r(p).

Closed sets

1. E\subset X,p\in X. We say p is a limit point of E if \forall r>0, (B_r(p)\cap E)\backslash {p}\neq \phi.

   Let E' be the set of limit points of E.

2. E is closed if E'\subset E

Example:

X=\mathbb{R}^2, E=[0,1)\times [0,1).

(1,1) is a limit point.

X=\mathbb{R},E=\{\frac{1}{n},n\in \mathbb{N}\}

0 is the only limit point. E'=\{0\}