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/0:
Oh crap! Makes sense now, thanks kamil

/0:
Is there an example where in a metric space (X,d) the distance between any two closed sets is 0 even if the sets are disjoint? I've been mulling over it for hours and I can't seem to think of an example!

kamil9876:
Rationals are ussually good examples of these kinds of not so pictorialy obvious properties.

I assume you mean that the distance between sets and is
Consider



That get's me thinking... what about complete metric spaces?

edit: nope, not even, consider the space:



The distance between these two disjoint sets is . It is also true that in this space a sequence is convergent iff it is evntually constant, and it is also true that in this space a sequence is cauchy iff it is eventually constant. Thus it's complete. The two sets are both closed and open (in fact, this space is totally disconnected: ie any subset of it is both closed and open).

(graph this to verify these properties easily).

/0:
Thanks kamil ;)

Why is it that instead of ?

Also, what does a ball in the French Metro metric look like



I tried to graph it but it seems so messy, there's so many different cases for 't' and restrictions etc.

humph:

--- Quote from: /0 on April 10, 2010, 12:45:56 am ---Thanks kamil ;)

Why is it that instead of ?

--- End quote ---
Counterexample (this is a pretty standard one too - just think of any function that isn't one-to-one):
given by . Take , so that . Then
.


--- Quote from: /0 on April 10, 2010, 12:45:56 am ---Also, what does a ball in the French Metro metric look like



I tried to graph it but it seems so messy, there's so many different cases for 't' and restrictions etc.

--- End quote ---
Think about what can be, and what happens for different values of .


Note that is a line segment, while is the Euclidean ball centred at of radius .

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