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Let (X,d) be a metric space. Let A c X.

What does it mean that A is of the first category?

What does it mean that A is of the first category?

A set of the first category is a countable union of rare sets.

(The closure of a rare set contains no ball.)

(Every rare set is of the first category.)

a set of the first category = a meager set

(The closure of a rare set contains no ball.)

(Every rare set is of the first category.)

a set of the first category = a meager set

Let (X,d) be a metric space. A c B c X.

If B is meager in X, then A is meager in X.

Decide if it's true. Prove your answer.

If B is meager in X, then A is meager in X.

Decide if it's true. Prove your answer.

TRUE

Let (X,d) be a metric space. A c B c X.

If A is meager in X, then B is meager in X.

Decide if it's true. Prove your answer.

If A is meager in X, then B is meager in X.

Decide if it's true. Prove your answer.

FALSE.

Let (X,d) be a metric space. A c B c X.

If A is rare in B, then A is rare in X.

Decide if it's true. Prove your answer.

If A is rare in B, then A is rare in X.

Decide if it's true. Prove your answer.

TRUE

To prove this, use (p->q) <== (-q -> -p).

page 178 in the palace notebook

To prove this, use (p->q) <== (-q -> -p).

page 178 in the palace notebook

Let (X,d) be a metric space. A c B c X.

If A is rare in X, then A is rare in B.

Decide if it's true. Prove your answer.

If A is rare in X, then A is rare in B.

Decide if it's true. Prove your answer.

FALSE.

Let X = |R, and let A be the Cantor.

Then A is rare in X.

Let B=A, then AcB but A is not rare in B.

Let X = |R, and let A be the Cantor.

Then A is rare in X.

Let B=A, then AcB but A is not rare in B.

Let (X,d) be a metric space. A c B c X.

If A is meager in B, then A is meager in X.

Decide if it's true. Prove your answer.

If A is meager in B, then A is meager in X.

Decide if it's true. Prove your answer.

TRUE.

Use: If A is rare in B, then A is rare in X.

Use: If A is rare in B, then A is rare in X.

Let (X,d) be a metric space. A c B c X.

If A is meager in X, then A is meager in B.

Decide if it's true. Prove your answer.

If A is meager in X, then A is meager in B.

Decide if it's true. Prove your answer.

FALSE

Let A=B={1}. Let X=|R.

Then A is meager in X, but A is not meager in B.

Let A=B={1}. Let X=|R.

Then A is meager in X, but A is not meager in B.

Let f[n], g[n] be sequences of functions X->|C. Let f,g : X->|C.

Suppose that f[n] and g[n] converge uniformly to f and g respectively.

Suppose that both the sequences are uniformly bounded.

Is it true that f[n]*g[n] converges uniformly to f*g?

Suppose that f[n] and g[n] converge uniformly to f and g respectively.

Suppose that both the sequences are uniformly bounded.

Is it true that f[n]*g[n] converges uniformly to f*g?

YES

page 178 in the palace notebook

page 178 in the palace notebook

Let (X,d) be a bounded metric space.

Let H be the set of all closed subsets of X except the empty set.

Define h:HxH->R by h(A,B) = sup{|d(x,A) - d(x,B)| : x:-X}.

Prove that (H,h) is a metric space.

Let H be the set of all closed subsets of X except the empty set.

Define h:HxH->R by h(A,B) = sup{|d(x,A) - d(x,B)| : x:-X}.

Prove that (H,h) is a metric space.

This is called the Hausdorff metric space.

page 179 in the palace notebook

page 179 in the palace notebook

Let (X,d) be a bounded metric space.

Let H be the set of all closed subsets of X except the empty set.

What is the definition of the Hausdorff metric space?

(Just recall the definition without proving that it is a metric space.)

Let H be the set of all closed subsets of X except the empty set.

What is the definition of the Hausdorff metric space?

(Just recall the definition without proving that it is a metric space.)

h:HxH->R by h(A,B) = sup{|d(x,A) - d(x,B)| : x:-X}.

(H,h) is a metric space, see the previous item.

(H,h) is a metric space, see the previous item.