Re: Complete metric quotient spaces

William Elliot wrote:

On Tue, 12 Sep 2006, rusty wrote:

The usual way to define a quotient pseudo-metric is as the infimum
of the distances between points in two equivalence classes.

Yes I cancelled the message shortly after lunch yesterday (french time)
and you apparently just replied today to this cancelled mail.

It's been posted for a long time and deletes don't always take hold.

It was posted at 10:58 and cancelled at 14:07 french time.

In my haste I confused two results :

- For a compact metric space, the quotient will be a (compact) metric
space iff all equivalence classes are closed. (In general for the
quotient to be metrisable each equivalence class must be closed as the
inverse image of a closed set under a continuous mapping!)

What metric is that? Certainly not the metric described a bove.

Yes the quotient metric d(C_1,C_2) given as the infimum of d(x_1,x_2) where
x_1 runs over C_1 and x_2 over C_2. Since X is compact, there is no problem.


- The topology on a subset of a complete separable metric space can be
given by a complete metric iff the subset is a G_delta.

What's a complete metric?

I wrote "complete separable metric space". Complete and separable are
adjectives that qualify "metric space". Do you have a problem?
--
rusty
.

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