Re: link between topological and metric spaces

On Sun, 27 Nov 2005 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

No, a metric space is Hausdorff and if the space isn't Hausdorff,
then any coarser (subset of) topology, isn't Hausdorff.

> 2) exactly equal to

No, a metric space is 1st countable, Hausdorff
and perfectly and monotonically normal.

Any space that isn't 1st countable or not Hausdorff or not
normal in the strongest sense of normal as above, isn't metrizable.

> 3) a superset of
>
Yes, the discrete metric
d(x,y) = 1, x /= y
d(x,x) = 0

produces the discrete topology which is finer, (super set of)
all other topologies for X.

> ( 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.
>
I'm dubious as there seems little more to consider. One may ask if
there's a Hausdorff space without a coarser metric space. Other than
that the question seems completely settled.

.

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