Re: link between topological and metric spaces

In article <dmcl6b$mv1$1@xxxxxxxxxxxxxxxx>, <lhr_cool_guy@xxxxxxxxx>
wrote:

> Given a topological space ( X, t ), can we find a metric d s.t. ( X, d
> ) is a metric space and the topology induced by this metric space is:
>
> 1) a subset of
> 2) exactly equal to
> 3) a superset of
>
> ( X, t )? If yes, how, if not why not?
>
> If someone can point me to an appropriate book or research paper or
> give me some pointers about how to solve this, I will be much grateful.

3) is trivial: d(x,y) = 1 if x \ne y, 0 if x = y yields the discrete
topology (ALL sets are open).

1) can't always be done. If it can be done, the topology must be
Hausdorff. I doubt that this is sufficient, though.

2) Google "metrization theorems", especially the Smirnov-Nigata theorem.

--Ron Bruck

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