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Let A,B be subsets of a metric space. Let p be a point in this
space.

Suppose that for every ball with center p,

if this ball is contained in A, then it is not contained in B.

What can we conclude about point p?

Suppose that for every ball with center p,

if this ball is contained in A, then it is not contained in B.

What can we conclude about point p?

p does not belong to Int( A n B )

Let G be an open subset of a metric space X.

Show that for every A c X,

G n Clo(A) is non-empty => G n A is non-empty.

Show that for every A c X,

G n Clo(A) is non-empty => G n A is non-empty.

______

Let G be an open subset of a metric space X.

Show that for every A c X,

G n A is empty => G n Clo(A) is empty.

Show that for every A c X,

G n A is empty => G n Clo(A) is empty.

_____

Let f[n]:E->C, g[n]:E->C be functional sequences.

Suppose that f converges uniformly to the zero function.

Suppose that g[n] is uniformly bounded.

What can we conclude about f[n]*g[n]?

Suppose that f converges uniformly to the zero function.

Suppose that g[n] is uniformly bounded.

What can we conclude about f[n]*g[n]?

f[n]*g[n] converges uniformly to the zero function.

Consider the series x^n / (1 - x^n).

Where does it converge? Consider x complex, |x| != 1.

Where does it converge? Consider x complex, |x| != 1.

|x|!=1 => [ |x|<1 <=> converges ]

X set, M c X.

W = {E c X : M\E is finite}

Describe R(W) - the ring generated by W.

W = {E c X : M\E is finite}

Describe R(W) - the ring generated by W.

R(W) = {E c X : M\E is finite v MnE is finite}

X set, W c P(X).

F : W -> [-oo,oo]

What does it mean that F is subadditive?

F : W -> [-oo,oo]

What does it mean that F is subadditive?

/\A,B:-W [ AuB :- W ==> F(AuB) <= F(A) + F(B) ]

X set, W c P(X).

F : W -> [-oo,oo]

What does it mean that F is finitely subadditive?

F : W -> [-oo,oo]

What does it mean that F is finitely subadditive?

For every natural n, if A[1],...,A[n] belong to W, and if their
union belongs to W,

then F(their union) <= +(k=1 to k=n) [ F(A[k]) ].

then F(their union) <= +(k=1 to k=n) [ F(A[k]) ].

X set, W c P(X).

F : W -> [-oo,oo]

What does it mean that F is countably subadditive?

F : W -> [-oo,oo]

What does it mean that F is countably subadditive?

For every sequence A[n] of sets in W whose union is also in W,

we have F(their countable union) <= +(n=1 to n=oo) [ F(A[n]) ].

we have F(their countable union) <= +(n=1 to n=oo) [ F(A[n]) ].

X set.

What is "the counting measure on X"?

What is "the counting measure on X"?

F : P(X) -> [0,oo]

If A c X is finite, then F(A) = the number of elements of A.

If A c X is infinite, then F(A) = oo.

Notice that it is countably additive!

If A c X is finite, then F(A) = the number of elements of A.

If A c X is infinite, then F(A) = oo.

Notice that it is countably additive!