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William Elliot wrote:
>> By the way (0,1) is
>> a Polish space since it is homeomorphic to the complete metric space
>> (-\infty, \infty). I suggest you look up the references in Chapter IX of
>> Bourbaki's General Topology for your own eduaction. And no I do not mean
> a
>> closed G_delta. Please see my two other messages and please check
>> carefully before replying to a cancelled message.
>
> (0,1) is not a complete metric space by the definition
> every Cauchy sequence converges to a point.
> However with a topologically equivalent, but not uniformly equivalent
> metric, (0,1) can be mapped to R which is complete.
You gradually seem to be working out what a Polish space is ;-)
>> <news:4506cb3b$0$11850$636a55ce@xxxxxxxxxxxx>
>
> [Malformed "news" URL: Invalid newsgroup specified]
>
>> <news:4507c708$0$11852$636a55ce@xxxxxxxxxxxx>
>
> [Malformed "news" URL: Invalid newsgroup specified]
What else do you expect if you use Pine? The recommended way to access
messages like this from Pine is to enter the Message-Id in the message box
here:
http://groups.google.com/advanced_search
>
>> I have just noticed your strange comments and the extraordinary way
>> you have edited/rearranged/concatenated some of my previous posts. I
>> see you do not understand the definition of a Polish space. As for
>> algebras, have you heard of the Stone-Weierstrass theorem or the
>> Gelfand-Naimark correspondence?
>
> No.
>
>> My use of the term algebra is current amongst professional
>> mathematicians: I'm sorry if you were unaware of this.
>
> The uses of algebra I'm familiar with relate to vector spaces
> and measure theory.
L^\infty(X,m) forms a commutative algebra if (X,m) is a measure space.
The endomorphisms of a finite-dimensional vector space form a noncommutative
algebra.
>
>> Why did you delete the first reference to the result on quotients
>> of compact metric spaces from Bourbaki's General Topology?
>
> Beyond my reach.
>
>> As for your "no do have" concerning Bourbaki, if you are not near a
>> mathematics library or do not possess this book (as I was yesterday
>> afternoon), you can find all the relevant pages on google books by
>> an intelligent use of search terms within the book.
>
> No I'm not near a univerisy library and I've little success with google
> books unless your suggesting I buy a copy. That I can do thru Powell's
> Books and have Kelley's 'General Topology' and Steen's 'Counterexamples in
> Topology'.
I managed to find the two relevant pages in Bourbaki this way. First I
looked in the contents and noted on which pages the results were likely to
appear ("quotient spaces" and "Polish spaces" respectively). Then I entered
these as search terms and picked out the nearest page. Quite simple. No
need to buy the book. I too have Kelley but it's not a useful reference for
Polish spaces.
>> In addition I have checked your statement about cancelled messages
>> using two news readers on accounts outside France (pmms.cam.ac.uk
>> and math.berkeley.edu) and I am unable to confirm what you wrote.
>
> As you see, cancelled messages do get through.
Pine again ;(
By the way, how do you cancel messages in Pine?
>> I wish you luck with Bourbaki.
>
> Kelly claims a G_delta within a complete metric space is homeomophic to a
> complete metric space. Gives suggestions how to construct the
> homeomorphism.
The implications in both directions are proved in Bourbaki or Arveson (or
probably even in Kuratowski's Topologie).
G_delta's arise naturally as follows. A group acts on a convex set which
also has the structure of a compact metric space. The extreme points form a
G_delta on which the group acts. This is therefore equivalent to a group
action on a metric space that is complete and separable (i.e. a "complete
separable metric space"), in part the subject of this thread.
--
rusty
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