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.

The key word is "metrizable". See e.g.
<http://en.wikipedia.org/wiki/Metrizable>.
(3) is always true: take d(x,y) = 0 for x = y, 1 otherwise, so the
topology of (X,d) is discrete.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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