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Words 3741

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1. Set operations

A set is a collection of some items such as outcomes of an experiment.

We denote sets using upper case letters, say A and write a ∈ A if a is an element belonging to A.

If A and B are two sets, then the notation A ⊆ B means that the set A is included in B , i.e. each element of A is also an element of B :

A⊆B

iff [∀a ∈ A : a ∈ B ]

If A ⊆ B and A = B then we say that the set A forms a proper subset of B and write A ⊂ B .

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In what follows all sets will be subsets of a larger set Ω. The complement of A in Ω is denoted by Ac and represents elements of Ω which do not belong to A:

Ac = { ω ∈ Ω : ω ∈ A}

/

The complement of the set Ω is given by the empty set ∅.

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For any sets A ⊆ Ω, B ⊆ Ω, we denote by A ∪ B and A ∩ B their union and intersection. The union represents points which belong to A or B :

A ∪ B = {ω ∈ Ω : ω ∈ A or ω ∈ B } while intersection corresponds to points which belong to both sets

A ∩ B = {ω ∈ Ω : ω ∈ A and ω ∈ B }

If A and B are disjoint sets, i.e. A ∩ B = ∅, then their union will be denoted by A + B . Finally, the difference and the symmetric difference are deﬁned as

B − A = B ∩ Ac = {ω : ω ∈ B and ω ∈ A} − difference

/

A∆B = (A − B ) ∪ (B − A) − symmetric difference

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The operations of union and intersection are governed by certain laws.

They are given by

(i) identity laws:

A∪∅ = A

and

A∩Ω = A

(ii) domination laws:

A∪Ω=Ω

and

A∩∅=∅

A∪A = A

and

A∩A=A

A∪B =B∪A

and

A∩B =B∩A

(iii) idempotent laws

(iv) commutative laws:

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(v) associative laws:

A ∪ (B ∪ C ) = (A ∪ B ) ∪ C

and

A ∩ (B ∩ C ) = (A ∩ B ) ∩ C

(vi) distributive laws:

A ∩ (B ∪ C ) = (A ∩ B ) ∪ (A ∩ C )

and

A ∪ (B ∩ C ) = (A ∪ B ) ∩ (A ∪ C )

(vii) absorption laws

(A ∩ B ) ∪ B = B

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and

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A ∩ (A ∪ B ) = A

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Lastly, complementation is governed by

(viii) De Morgan laws: if A ⊆ B

then

(A ∩ B )c = Ac ∪ B c

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B c ⊆ Ac and (A ∪ B )c = Ac ∩ B c

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The operations of union and intersections are also deﬁned for arbitrary families of sets. Let T be an arbitrary index set, and suppose that At ⊆ Ω for each t ∈ T . Then

At = {ω ∈ Ω : ∃ t ∈ T ω ∈ At } t ∈T

At = {ω ∈ Ω : ∀ t ∈ T ω ∈ At } t ∈T

We have

(i)

t ∈T

At ⊆ At0 ⊆

t ∈T

At

for all

t0 ∈ T

(ii) if At ⊆ Bt for all t ∈ T then

At ⊆ t ∈T

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Bt

At ⊆

and

t ∈T

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t ∈T

Bt t ∈T

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(ii) De Morgan laws

At ]c =

[ t ∈T

Ac t and

t ∈T

[

At ]c =

t ∈T

Ac t t ∈T

(iv) associativity laws:

At ∪ t ∈T

[At ∪ Bt ] and

Bt = t ∈T

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t ∈T

At ∩ t ∈T

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[At ∩ Bt ]

Bt = t ∈T

t ∈T

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Cartesian products

If Ω and Ω′ are two sets then their Cartesian product Ω × Ω′ represents the collection of ordered pairs (ω, ω ′ ) such that ω ∈ Ω and ω ′ ∈ Ω′ . More generally, if A ⊆ Ω and B ⊆ Ω′ then

A × B = {(ω, ω ′ ) : ω ∈ A

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and

ω′ ∈ B}

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We have

A × B = ∅ iff

A=∅

or B = ∅

(A1 ∩ A2 ) × (B1 ∩ B2 ) = A1 × B1 ∩ A2 × B2

[A × B ]c = Ac × Ω2 ∪ Ω1 × B c

A × (B1 − B2 ) = A × B1 − A × B2

(A1 − A2 ) × B = A1 × B − A2 × B

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A×

A × Bt

Bt = t ∈T

A×

t ∈T

A × Bt

Bt = t ∈T

t ∈T

At × B = t ∈T

At × B t ∈T

At × B = t ∈T

At × B t ∈T

Finally if A1 ⊆ A2

and

B1 ⊆ B2

then

A1 × B1 ⊆ A2 × B2

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2. Extended real line

The extended real line, denoted by R , consists of all real numbers with added ∞ and −∞.

The addition and multiplication of ﬁnite reals is deﬁned in the usual way.

We also make the following conventions. For any ﬁnite real a

(i )

a+∞=∞+∞=∞ a − ∞ = (−∞) − ∞ = −(∞) + (−∞) = −∞ − (−∞) = ∞

(ii )

if

a > 0 then

a · ∞ = ∞ · ∞ = (−∞) · (−∞) = (−a)(−∞) = ∞ a · (−∞) = (−∞) · ∞ = ∞(−∞) = (−a)∞ = −∞

(iii )

0 · ∞ = 0 = 0 · (−∞)

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(iv )

a a =

=0

∞

−∞

The remaining operations (such as ∞ − ∞ or

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∞

∞)

are undeﬁned.

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If A is a subset of reals then the least upper bound of A is denoted by sup A and is deﬁned as the smallest extended real number a such that x ≤ a for all x ∈ A. Thus

∀x ∈ A : x ≤ a if [∀x ∈ A : x ≤ b ] then

a≤b

Similarly, the greatest lower bound of A is denoted by inf A and it represents the largest number c such that c ≤ x for all x ∈ A. Thus

∀x ∈ A : c ≤ x if Dorota M. Dabrowska (UCLA)

[∀x ∈ A : b ≤ x ] then

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b≤c

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If A represents a sequence of numbers A = {xn : n ≥ 1} then we write sup A = supn xn and inf A = infn xn . We further deﬁne the upper and the lower limits of the sequence {xn : n ≥ 1} by setting lim inf = sup inf xn n k

n ≥k

lim sup = inf sup xn n k n ≥k

By deﬁnition

−∞ ≤ lim inf xn ≤ lim sup xn ≤ ∞ n n

If lim infn xn = lim supn xn then the common value is denoted by lim xn and we say that the sequence {xn : n ≥ 1} has a limit. In particular, if

{xn : n ≥ 1} is monotone then its limit exists (in the extended real line).

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We further note that the sequence bk = inf xn n ≥k

is increasing. Therefore its limit exists and is equal to supk bk . Thus lim inf xn = lim ( inf xn ). n k →∞ n≥k

Similarly, ck = supn≥k xn forms a decreasing sequence so it has a limit equal to infk ck and we have lim sup xn = lim sup xn . n Dorota M. Dabrowska (UCLA)

k →∞ n≥k

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3. Sequences of sets For any A ⊆ Ω, its indicator function is

1A (ω ) = 1 if

ω∈A

= 0 if

ω∈A

We have

1A∪B = max{1A , 1B }

1A∩B = min{1A , 1B } = 1A 1B

1Ac

= 1 − 1A

1A∆B = |1A − 1B |

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Similarly, if A1 , ...., An , ..... is a sequence of subsets of Ω then

∞

ω∈

An

iff

n ≥1

n ≥1

∞

ω∈

An

iff

n ≥1

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sup 1An (ω ) = 1

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inf 1An (ω ) = 1

n ≥1

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We can use this correspondence to deﬁne limit superior and limit inferior of a sequence of sets.

Deﬁnition Let {An : n ≥ 1} and A be subsets of Ω.

(a) The set

∞

∞

lim sup An = n An k =1 n =k

is the limit superior of the sequence {An : n ≥ 1} In other words, ω ∈ An for inﬁnitely many values of n. We write lim supn An = {An i .o .} and we say that An occurs inﬁnitely often.

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(b) The set

∞

∞

lim inf An = n An k =1 n =k

is the limit inferior of the sequence {An : n ≥ 1}. In this case ω belongs to An for all but ﬁnitely many values of n. We shall often denote this set as lim infn An = {An ev .} and we say that An occurs eventually. (c) The sequence {An : n ≥ 1} converges to A if lim sup An = lim inf An = A n n

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In terms of characteristic functions

B = lim sup An

iff

1B = lim sup 1An

C = lim inf An

iff

1C = lim inf 1An

n

n

n

n

This implies in particular, lim inf An ⊆ lim sup An n n

We can also verify this directly. For any k ≥ 1 set

Ck =

An

and

Bk =

n ≥k

An n ≥k

We have

Ck ⊆ Ak ⊆ Bk

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for all k . If ω ∈ Ck for some k , then k ω∈

Bi i =1

because B1 ⊇ B2 ⊇ . . . is a decreasing sequence. Moreover,

C1 ⊆ C2 . . . ⊆ ... is an increasing sequence so that ω ∈ Ck implies ω ∈ Bi for all i ≥ k .

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For handling increasing and decreasing sequences of sets, we have the following proposition, paralleling properties of monotone sequence of numbers. Proposition Let A1 , A2 , . . . be subsets of Ω. Then

(a) If A1 ⊆ A2 . . . is an increasing family of sets then

An → A = ∞ 1 An . (this is denoted by An ↑ A). n= (b) If A1 ⊇ A2 . . . is a decreasing family of sets then

An → A = ∞ 1 An . (this is denoted by An ↓ A). n= Dorota M. Dabrowska (UCLA)

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Proof Let A =

∞ n =1 A n .

a) Since A1 ⊆ A2 ⊆ ...., we have

∞

∞

An = Ak

An = A and n =k

n =k

Therefore

∞

A=A

lim sup An = n k =1

∞

lim inf An = n Ak = A k =1

Part (b) follows by complementation.

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Disjointiﬁcation trick

Let A1 , ...An , .... be an arbitrary sequence of sets, An ⊆ Ω. Then

∞

∞

An = n =1

Bn

(1.1)

n =1

for sets n −1

Bn = An −

Ai , B1 = A1 i =1

We have

(a) Bn ∩ Bm = ∅ if n = m;

(b)

k n =1 A n

=

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k n =1 B n

for any k ≥ 1

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Then (b) implies (1.1) because k k

An =

Ck =

Bn n =1

n =1

is an increasing sequence of sets

C1 ⊆ C2 ⊆ ....Cn ⊆ .... and the union is equal to

∞

∞

Bi = C

Ai = i =1

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i =1

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4. Metric spaces

Deﬁnition If X is a set and d : X × X → R+ is a function satisfying

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

(ii) d (x , z ) ≤ d (x , y ) + d (y , z ) (triangle inequality);

(iii) d (x , x ) = 0. then d is called a semi-metric. If (iii) is replaced by

(iii’) d (x , y ) = 0 iff x = y then d is called a metric. The pair (X , d ) is correspondingly referred to as a semi-metric or metric space.

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The diameter of a set A of metric space X is deﬁned diam (A) = sup{d (x , y ) : x ∈ A, y ∈ A}

An open ball centered at x ∈ X and having radius ε is the set of points y ∈ X whose distance from X is less than ε

B (x , ε) = {y ∈ X : d (x , y ) < ε}

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A subset K is bounded if there exists a ball B (of a ﬁnite radius) such that

K ⊂ B.

A subset K is totally bounded if for every ε > 0 it can be covered by ﬁnitely many balls of radius ε.

Every totally bounded set is bounded, however, a bounded set need not be totally bounded.

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Example Let X be an arbitrary set and deﬁne d (x , y ) = 1 x = y

= 0 x =y

Then d is a metric, usually referred to as the discrete metric. A ball centered at x ∈ X is

B (x , ε) = {x }

=X

if ε ≤ 1 if ε > 1

Thus X is bounded. However, if X is inﬁnite, then it cannot be covered by means of a ﬁnite number of balls of radius ε ≤ 1.

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Example

Let X = R be equipped with the usual metric d (x , y ) = |x − y |.

Then each ball is of the form (x − ε, x + ε), ε > 0.

The set of reals is unbounded hence not totally bounded.

On the other hand, ﬁnite intervals [a, b ] or (a, b ), −∞ < a < b < ∞ are both bounded and totally bounded.

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We say that a sequence of points {xn : n ≥ 1} ⊆ X converges to x , x ∈ X if d (xn , x ) → 0. We write in this case xn → x or lim xn = x . Thus a sequence {xn } converges to x iff every ball centered at x contains all but a ﬁnite number of terms of the sequence {xn }.

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Every convergent sequence is bounded and totally bounded.

In addition, any convergent sequence of a metric space satisﬁes the

Cauchy condition, i.e. for every ε > 0 there exists N = N (ε) such that d (xn , xm ) < ε for all n, m ≥ N .

A metric space is called complete if the converse holds as well,

i.e. every Cauchy sequence has a limit in X .

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Example This deﬁnition of convergence coincides with the usual deﬁnition of convergence of sequences of real numbers in X = R with metric d (x , y ) = |x − y |. Every Cauchy sequence of reals forms a convergent sequence, therefore X = R is a complete metric space with respect to d . However, if we take X = (−1, 1) with the same metric, then the sequence {1 − 1/n : n ≥ 2} satisﬁes the Cauchy condition but has no limit in X so that (X , d ) is not complete.

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A common example of metric spaces is provided by vector spaces V equipped with a norm.

Deﬁnition Let V be a vector space. A function · : V → R+ satisfying

(i) x + y ≤ x + y (triangle inequality);

(ii) αx = |α| x for any scalar α

(iii) if x = 0 then x = 0 is called a semi-norm. If instead of (iii), the function · satisﬁes

(iii’)

x =0

iff x = 0

then · is called a norm. The pair (V , · ) is called a semi-normed or a normed space, respectively.

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A norm · induces a metric on the vector space V . For any x , y ∈ V we deﬁne it as d (x , y ) = x − y

If the resulting metric space is complete then the pair (V , · ) is called a

Banach space.

Example In the case of X = R k , we can choose for example norms k k

x

1

|xi |

= i =1

x

2

xi2

= i =1

x

∞

= max |xi | i =1,...,k

Here · 1 corresponds to the ℓ1 norm, · 2 corresponds to the Euclidean norm and · ∞ to the maximal norm. For each if these choices, the induced metric d (x , y ) turns R k into a complete metric space.

Dorota M. Dabrowska (UCLA)

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Note that the three norms satisfy x ∞ ≤ x 1, x 2≤k x

1

x 1≤ x 2≤k x 1 k ∞

so that they can be viewed as equivalent. More generally, if ·

·

2

1

and

are two norms in a vector space V then we call them equivalent if

there exist constants a > 0, b > 0 such that x 1

≤a x

2

x

2

≤b x

1

for any x ∈ V .

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5. Topological spaces

A topological space is a set X equipped with an operation of closure or the operation of interior.

The closure operation assigns to an arbitrary subset A ⊆ X a set A ⊆ X such that

(F-i) ∅ = ∅

(F-ii) A ⊆ A

(F-iii) A = A

(F-iv) A ∪ B = A ∪ B

The set A is called closed if A = A.

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From this deﬁnition it follows that a ﬁnite union of closed sets is also closed. Moreover,

(i) if A ⊆ B then A ⊆ B

(ii) A − B ⊆ A − B

(iii) if {At : t ∈ T } is an arbitrary collection of subsets of X then

At ⊆ t ∈T

At ⊆ t ∈T

At ⊆ t ∈T

At t ∈T

(iv) X = X

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If F is the collection of closed sets in X then the preceding implies

(F’-i) ∅ ∈ F , X ∈ F

(F’-ii) intersection of an arbitrary number of closed sets is also closed

(F’-iii) union of a ﬁnite number of closed sets is closed.

In particular, if A ⊂ X then its closure A is the smallest closed set containing A: we have

A=

Dorota M. Dabrowska (UCLA)

{F ∈ F : A ⊂ F }

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A set A is called open if its complement Ac = X − A is closed. If we denote by G the collection of all open sets in X then

(G’-i) ∅ ∈ G , X ∈ G

(G’-ii) union of an arbitrary number of open sets is also open

(G’-iii) intersection of a ﬁnite number of open sets is open.

If A ⊂ X then its interior int A is the largest open set contained in A: we have int A = {G ∈ G : G ⊂ A}

A point x is in the interior of A iff there exists and open set G such that x ∈G

and

G ∩ (X − A) = ∅

Moreover, int A = X − (X − A)

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Alternatively, the interior operation assigns to every set A of X a subset such that

(G-i) int X = X

(G-ii) int A ⊆ A

(G-iii) int int A = int A

(G-iv) int A ∩ B = int A ∩ int B

The set A is called open if A = int A. Its closure is given by

A = X − int (X − A).

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The boundary of a set A is deﬁned as δA = A − int A

The boundary is a closed set and contains points of both A and its complement. Dorota M. Dabrowska (UCLA)

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In many circumstances, the topology of a space can be deﬁned by specifying its base. A family B of open subsets of X is called a base, if every open set of X is a union of some sets belonging to B .

A base always exists because we can take it equal to G .

However, the base can be often chosen as a smaller collection of open sets.

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We can now specialize these concepts to metric spaces. If X is a metric space with metric d , then we can deﬁne closure of a set A as the collection of points which are limits of sequences contained in A:

A = {x ∈ X : x = lim xn

for some sequence

{xn }∞ 1 ⊆ A}. n= Theorem

(i) We have x ∈ A x ∈B

iff B ∩ A = ∅ for every open ball B such that

(ii) A set is open iff it is a union of some open balls.

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Separable topological and metric spaces

A set A in a topological space X is called dense if A = X . Alternatively, a set A is dense iff every nonempty open set of X contains points of A.

A topological space is called separable if it has a countable dense set.

In particular, every X with a countable base is separable.

In the case of metric spaces, these two concepts are equivalent.

Separability of a metric space is also equivalent to the assumption that each discrete subspace is at most countable.

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Example

If X = R with its usual metric d (x , y ) = |x − y | then the set of rationals is dense in R . The countable base of X corresponds to all balls centered at rationals and having a rational radius. The set of irrational numbers,

IR = R − Q is also dense in R because every ball B (x , ε) contains irrational numbers.

Dorota M. Dabrowska (UCLA)

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Example Bounded functions.

If I is an arbitrary set, then ℓ∞ (I ) is the space of all real valued bounded functions x : I → R endowed with supremum norm x ∞

= sup |x (t )| t ∈I

This is a Banach space. It is separable iff I is ﬁnite. If I is a ﬁnite set then the collection of all scalar functions assuming rational values is a countable dense set of ℓ∞ (I ). On the other hand if I is inﬁnite, then characteristic functions 1A of sets A ⊂ I form an uncountable collection of functions satisfying 1A − 1B ∞ = 1. Therefore, the corresponding

ℓ∞ (I ) space is non-separable.

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Example

The space C = C ([0, 1]k ) of real valued continuous functions on [0, 1]d equipped with uniform metric x −y

∞

= sup |x (t ) − y (t )| t ∈[0,1]k

is separable and complete.

In particular, for k = 1, the collection of piecewise linear functions with

“kinks” located at rational points form a countable dense set in C ([0, 1]).

For k > 1, the separant can be chosen in a similar fashion.

Dorota M. Dabrowska (UCLA)

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...Network Hardening Assignment 8 The Internet is vulnerable to myriads kinds of attacks, due to: 1. Vulnerabilities in the TCP-IP protocol 2. No global flow control mechanisms The above two problems lead to many TCP exploits and the dreaded DDoS attacks. We have devised a method of incrementally upgrading the network infrastructure at the transport level that solves the above problems and makes the network significantly more resilient to attacks, particularly the DDoS attack. The approach uses "hardened routers" -- routers that can do simple cryptographic functions (encryption, signatures) on all packets flowing int he network, as well as to participate in a hierarchical control network. We show how incremental deployment of such routers can make the Internet safer. Like all things dynamic, change is inevitable. Such is the case with your network environment. Upgrades and modifications to the network architecture can sometimes expose (or create) security holes. As such, it is important to consistently evaluate the Making a Business Case for Network Hardening Hardening a network does not always translate into spending large quantities of money. However, money will be required in some form or fashion. Whether that means spending it on new hardware, software, or man hours really depends on what needs to be addressed. It may include all of the above. The time may come when a cost/benefit analysis will be required by those in charge before hardening activities can move......

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...contribute, please send e-mail to stephen@sans.edu 1. Overview Martin’s Inc. purpose for this ethics policy is to establish a culture of openness, trust and integrity in business practices. Effective ethics is a team effort involving the participation and support of every Martin’s Inc. employee. All employees should familiarize themselves with the ethics guidelines that follow this introduction. Martin’s Inc. is committed to protecting employees, partners, vendors and the company from illegal or damaging actions by individuals, either knowingly or unknowingly. When Martin’s Inc. addresses issues proactively and uses correct judgment, it will help set us apart from competitors. Martin’s Inc. will not tolerate any wrongdoing or impropriety at anytime. Martin’s Inc. will take the appropriate measures act quickly in correcting the issue if the ethical code is broken. Any infractions of this code of ethics will not be tolerated. 2. Purpose Our purpose for authoring a publication on ethics is to emphasize the employee’s and consumer’s expectation to be treated to fair business practices. This policy will serve to guide business behavior to ensure ethical conduct. 3. Scope This policy applies to employees, contractors, consultants, temporaries, and other workers at Martin’s Inc., including all personnel affiliated with third parties. 4. Policy 1. Executive Commitment to Ethics 1. Top brass within Martin’s Inc. must set a......

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