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Let (X,d) be a metric space. Let (Y,g) be a complete metric
space. Let A be a dense subset of X. Consider a Lipschitz
function f:A->Y. Can this function be extended to f:X->Y and
still be Lipschitz?

YES

Prove that for every metric space (X,d) there exist:

(1) a complete metric space (X',d')

(2) an isometry f:X->X'

such that f(X) is dense in X'.

(1) a complete metric space (X',d')

(2) an isometry f:X->X'

such that f(X) is dense in X'.

page 53 in golden gate

or

page 138 in the book "Topological Spaces - From Distance to Neighborhood"

or

page 138 in the book "Topological Spaces - From Distance to Neighborhood"

Let {A[n]} be a sequence of sets. Define the lower limit of
this sequence.

\\//(k=1 to k=oo) //\\ (n=k to n=oo) A[n]

x belongs to the lower limit of a sequence of sets

iff

x belongs to almost all sets from this sequence

x belongs to the lower limit of a sequence of sets

iff

x belongs to almost all sets from this sequence

Let {A[n]} be a sequence of sets. Define the upper limit of
this sequence.

//\\(k=1 to k=oo) \\// (n=k to n=oo) A[n]

x belongs to the upper limit of a sequence of sets

iff

x belongs to infinitely many sets from this sequence

x belongs to the upper limit of a sequence of sets

iff

x belongs to infinitely many sets from this sequence

What does it mean that a sequence of sets converges?

The upper limit of this sequence is contained in the lower limit.

For all x, if x belongs to infinitely many sets of this sequence, then it belongs to almost all sets of this sequence.

For all x, if x belongs to infinitely many sets of this sequence, then it belongs to almost all sets of this sequence.

What does it mean that a sequence of sets is increasing?

{A[n]} sequence of sets

/\n:-N [ A[n] c A[n+1] ]

(A constant sequence is increasing.)

/\n:-N [ A[n] c A[n+1] ]

(A constant sequence is increasing.)

What does it mean that a sequence of sets is decreasing?

{A[n]} sequence of sets

/\n:-|N [ A[n+1] c A[n] ]

(A constant sequence is decreasing.)

/\n:-|N [ A[n+1] c A[n] ]

(A constant sequence is decreasing.)

What does it mean that a sequence of sets is monotone?

['mo n.. tOun]

['mo n.. tOun]

It's increasing or it's decreasing.

See the previous two items.

See the previous two items.

A + B = (A\B) u (B\A)

What do you call this operation on sets?

What do you call this operation on sets?

the symmetric difference of two sets A and B

[si 'met rik]

[si 'met rik]

What is a disjoint class of sets?

Every two sets of this class are disjoint.

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2001.02.06

This definition is stupid.

==========

2001.02.06

This definition is stupid.