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Give the function which shows that the Cantor set is uncountable.

I = [0,1]. If J = [a,b], let J(0) = [a , a+(b-a)/3], let J(1) =
[b-(b-a)/3 , b].

If AcN and n:-N, let I(A(n)) = I(1A(1), 1A(2), ..., 1A(n)).

If n:-N then let C[n] = U{I(A(n)) : AcN}. Let's define C = //\\n:N [ C[n] ].

For AcN, let F(A) be the one point belonging to //\\n:-N I(A(n)).

If AcN and n:-N, let I(A(n)) = I(1A(1), 1A(2), ..., 1A(n)).

If n:-N then let C[n] = U{I(A(n)) : AcN}. Let's define C = //\\n:N [ C[n] ].

For AcN, let F(A) be the one point belonging to //\\n:-N I(A(n)).

Give a description of the Cantor set using series.

1) +(n=1 to oo) [ 2 * 1A(n) * 3^(-n) ]

For every Ac|N, the sum of the series above

uniquely determines an element of the Cantor set.

2) +(n=1 to oo) [ a[n] * 3^(-n) ]

Where a[n] is any sequence such that /\n:-|N [ a[n] :- {0,2} ].

For every Ac|N, the sum of the series above

uniquely determines an element of the Cantor set.

2) +(n=1 to oo) [ a[n] * 3^(-n) ]

Where a[n] is any sequence such that /\n:-|N [ a[n] :- {0,2} ].

Consider the metric space (P(|N),d), d(A,B) = 1/min(A+B).

Let {A[n]} be a sequence contained in P(|N).

(1) lim d(A[n],A) = 0

(2) lim A[n] = A (the lower and upper limits are both equal to A)

Show that (1) <=> (2).

Let {A[n]} be a sequence contained in P(|N).

(1) lim d(A[n],A) = 0

(2) lim A[n] = A (the lower and upper limits are both equal to A)

Show that (1) <=> (2).

page 169 in the palace notebook

Show that every function f:X->|R is the limit of some uniformly
convergent sequence of functions f[n]:X->|R.

f[n] = 1/n * [ n*f(x) ]

[r] denotes the integer part of r:-|R.

r-1 <= [r] <= r

[r] denotes the integer part of r:-|R.

r-1 <= [r] <= r

Can a countable dense subset of |R be written as a countable
intersection of open sets?

NO.

Argue by contradiction.

Use the Baire Category Theorem.

page 48 in OLDTIMER

Argue by contradiction.

Use the Baire Category Theorem.

page 48 in OLDTIMER

Let M be a countable dense subset of |R. Does there a exist a
sequence of continuous functions on |R which converges
pointwise to the characteristic function of M?

NO.

Let there be such a sequence {f[m]}. Put U[m] = f[m]-1 ((1/2,oo)). U[m] is the inverse image of (1/2,oo) under f[m]. U[m] is open because f[m] is continuous. Notice that

M = //\\n:-N \\//(m=n to m=oo) U[m].

M is written as the intersection of countably many open subsets of R. This is impossible. Contradiction. (see the previous item)

Let there be such a sequence {f[m]}. Put U[m] = f[m]-1 ((1/2,oo)). U[m] is the inverse image of (1/2,oo) under f[m]. U[m] is open because f[m] is continuous. Notice that

M = //\\n:-N \\//(m=n to m=oo) U[m].

M is written as the intersection of countably many open subsets of R. This is impossible. Contradiction. (see the previous item)

Let (X,d1) and (X,d2) be metric spaces.

What does it mean that d1 and d2 are similar metrics?

What does it mean that d1 and d2 are similar metrics?

/\x:-X /\{x[n]}cX [ lim d1(x[n],x)=0 <=> lim d2(x[n],x)=0 ]

Let z be a complex number.

Express differently:

|z|*|z| = ?

Express differently:

|z|*|z| = ?

|z|*|z| = z*conjugate(z)

Let X = (0,oo). Define d(x,y) = |x-y| / (x*y).

Show that (X,d) is a metric space.

Show that (X,d) is a metric space.

d(x,y) = |1/x - 1/y|

page 173 in palace

page 173 in palace

Let X = (0,oo). Define d(x,y) = |x-y| / (x*y). (X,d) is a
metric space.

Is is complete?

Is is complete?

NO.

{n} is a Cauchy sequence. It does not converge.

{n} is a Cauchy sequence. It does not converge.