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Let X be a complete metric space, with no isolated points. Let
A be a dense subset of X. Suppose that A is a countable
intersection of open sets. What can we conclude about A?

A is uncountable

page 48 in OLDTIMER

page 48 in OLDTIMER

Let X be a metric space with no isolated points. Let A be a
countable dense subset of X. Suppose that A is a countable
intersection of open sets. What can we conclude about the
metric space X?

X is incomplete

page 48 in OLDTIMER

page 48 in OLDTIMER

Let X be a complete metric space, with no isolated points. Let
A be a countable subset of X, which is a countable intersection
of open sets. What can we conclude about A?

A is not dense

page 48 in OLDTIMER

page 48 in OLDTIMER

Let X be a complete metric space. Let A be an infinitely
countable dense subset of X, which is a countable intersection
of open sets. Is this setup possible?

It is possible. For example:

Let X be {0} u {1/n : n:-N}.

Let A = X.

Let A be a countable intersection of itself.

Or, let A = X\{0}.

Let X be {0} u {1/n : n:-N}.

Let A = X.

Let A be a countable intersection of itself.

Or, let A = X\{0}.

Let X be a complete metric space with no isolated points. Let A
be a countable dense subset of X, which is a countable
intersection of open sets. Is this setup possible?

NO.

page 48 in OLDTIMER

page 48 in OLDTIMER

inf {x + 1/x : x>0} = ???

inf {x + 1/x : x>0} = 2

X set, W c P(X).

(1) X :- W

(2) A,B :- W => A u B :- W

(3) A :- W => X\A :- W

Prove that

(4) A,B :- W => A n B :- W

(1) X :- W

(2) A,B :- W => A u B :- W

(3) A :- W => X\A :- W

Prove that

(4) A,B :- W => A n B :- W

X \ (A n B) = (X\A) u (X\B) belongs, hence (4) is OK.

A \ B = A n (X\B) belongs, hence (5) is OK.

A \ B = A n (X\B) belongs, hence (5) is OK.

Let f be a complex function defined on an open set. Suppose
that this function is diffable at a point. Prove that at this
point function Re(f) has a partial derivative with respect to
the first variable.

page 18 in OLDTIMER

Let f be a complex function defined on an open set. Suppose
that this function is diffable at a point. Prove that at this
point function Im(f) has a partial derivative with respect to
the second variable.

page 18 in OLDTIMER

Let m be a natural number. Let z be a complex number with |z| <
1.

Express differently:

+(n=0 to n=oo) [ z^n * (n+m)! / n!*m! ] = ???

Express differently:

+(n=0 to n=oo) [ z^n * (n+m)! / n!*m! ] = ???

= 1 / (1-z)^(m+1)

page 190 in the palace notebook

page 190 in the palace notebook