Re: Questions about completeness

On Fri, 10 Jun 2005, Stephen J. Herschkorn wrote:

> We are told that completeness is a metric, not topological, property.
> What is an example of a topological space where two metrics induce the
> same topology but one metric space is complete and the other not?
> Clearly, the topological space cannot be compact.
>
Use 1/x homeomorphism between (0,1] and [1,oo), IIRC
and back track to a new metric D(x,y) = d(1/x, 1/y) for (0,1]

Also shows totally bounded isn't topological.

> The remaining questions address the relationship between
> order-completeness and metric-completeness.
>
> A bounded open interval of reals under the usual ordering and metric
> provides an example of a complete totally ordered set with a noncomplete
> metric inducing the order topology.
>
It's an example of a bounded complete, complete within bounds or
Dedekind complete linear order.

An example of a complete linear order is a closed interval.

> Consider a totally ordered set X whose order topology is metrizable.
> 1) If X is order-complete, does there necessarily exist a complete
> metric on X compatible with the order topology? (If X is not
> compact, then X cannot be totally bounded under such a metric.)

> 2) If there exists a complete metric which induces the order
> topology, is X necessarily order-complete?
>
How does the metric (S,d) induce the order topology (S,<=) ?
.

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