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Suppose that 0 < x < y.

x/y ??? (x+1)/(y+1)

x/y ??? (x+1)/(y+1)

x/y < (x+1)/(y+1)

Let X be a set. Let d : XxX -> [0,oo].

(1) /\x,y,z:-X d(x,z) <= d(x,y) + d(z,y)

Let u:-X, 0 < r <= oo.

Let U = {x:-X : d(u,x) < r}.

What can we conclude about set U?

(1) /\x,y,z:-X d(x,z) <= d(x,y) + d(z,y)

Let u:-X, 0 < r <= oo.

Let U = {x:-X : d(u,x) < r}.

What can we conclude about set U?

/\a:-U \/e>0 /\x:-X d(x,a)<e => x:-U

page 69 in gen top

page 69 in gen top

Let K be a set and /\k:-K d[k] : XxX -> [0,oo].

Let W = { U c X : \/k:-K /\a:-U \/e>0 /\x:-X d[k](x,a)<e => x:-U }.

Let x:T->X be a net in X, and let u:-X.

(1) /\k:-K /\e>0 \/p:-T /\t>p d[k](x[t] , u) < e

What can we conclude?

Let W = { U c X : \/k:-K /\a:-U \/e>0 /\x:-X d[k](x,a)<e => x:-U }.

Let x:T->X be a net in X, and let u:-X.

(1) /\k:-K /\e>0 \/p:-T /\t>p d[k](x[t] , u) < e

What can we conclude?

net x W-converges to u

page 71 in gen top

page 71 in gen top

Let K be a set and /\k:-K d[k] : XxX -> [0,oo].

(1) /\k:-K /\x:-X d[k](x,x)=0

(2) /\k:-K /\x,y:-X d[k](x,y) = d[k](y,x)

(3) /\k:-K /\x,y,z:-X d(x,z) <= d(x,y) + d(x,y)

Let W = { U c X : \/k:-K /\a:-U \/e>0 /\x:-X d[k](x,a)<e => x:-U }.

Let x:T->X be a net in X, which converges to u:-X.

(1) /\k:-K /\x:-X d[k](x,x)=0

(2) /\k:-K /\x,y:-X d[k](x,y) = d[k](y,x)

(3) /\k:-K /\x,y,z:-X d(x,z) <= d(x,y) + d(x,y)

Let W = { U c X : \/k:-K /\a:-U \/e>0 /\x:-X d[k](x,a)<e => x:-U }.

Let x:T->X be a net in X, which converges to u:-X.

page 71 in gen top

Let W be a s-ring in X.

Let y : W -> [0,oo] countably additive, y(O)=0.

In this context, what are the y-zero sets?

The symbol for the collection of all such sets,

let it be N(y).

Let y : W -> [0,oo] countably additive, y(O)=0.

In this context, what are the y-zero sets?

The symbol for the collection of all such sets,

let it be N(y).

N(y) = { EcX : \/A:-W EcA and y(A)=0 }

Recall that it is a s-ideal.

That is the subject matter of another item.

page 72 in 2nd measure

Recall that it is a s-ideal.

That is the subject matter of another item.

page 72 in 2nd measure

Imagine that a rectangle is written as a finite union of
rectangles, which can overlap only along the edges. Decide if
it must be true: The union of some two of these rectangles
forms a rectangle.

FALSE

ADD

AOC

BBC

ADD

AOC

BBC

Let f be a real-valued function defined on a connected subset
of |R.

Suppose that it is differentiable and that its derivative is bounded.

What can we conclude about this function?

Suppose that it is differentiable and that its derivative is bounded.

What can we conclude about this function?

It satisfies a Lipschitz condition.

Use Lagrange's mean value theorem to prove it.

page 156 in golden gate

Use Lagrange's mean value theorem to prove it.

page 156 in golden gate

Let f be a real-valued function defined on an open subset of |R.

Suppose that it is differentiable and Lipschitz.

What can we conclude about its derivative?

Suppose that it is differentiable and Lipschitz.

What can we conclude about its derivative?

Its derivative is bounded.

Notice that f may be defined on a disconnected subset of |R.

page 156 in golden gate

Notice that f may be defined on a disconnected subset of |R.

page 156 in golden gate

Let f be a real-valued function defined on an open subset of |R.

Suppose that it is differentiable and that its derivative is bounded.

Does this function have to be Lipschitz?

Suppose that it is differentiable and that its derivative is bounded.

Does this function have to be Lipschitz?

If it's defined on a connected subset, then yes.

But generally, no.

See page 80 in OLDTIMER for an example.

But generally, no.

See page 80 in OLDTIMER for an example.

Let R be a ring of sets in X.

Let W = { EcX : E:-R or X\E:-R }.

What can we conclude about W?

Let W = { EcX : E:-R or X\E:-R }.

What can we conclude about W?

W is an algebra of sets in X

and W = K(R)

page 67 in 2nd measure

and W = K(R)

page 67 in 2nd measure