It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with respect to inclusion) closed balls of vanishing diameters have non-empty intersection. This is usually referred to as Cantor's Intersection Theorem.
My question basically is:
Is it true that every completely metrizable topological space admits one compatible metric such that every intersection of nested closed balls has a non-empty intersection?
I will give some more details:
Let's call spherically complete a metric as above where every intersection of nested closed balls has non-empty intersection.
I'm very aware that not every complete metric is spherically complete. Only on this forum, there are several discussions on similar topics (see e.g. this question and the list of its related questions).
However, my question is not about a fixed metric, but about the existence of one single compatible metric for a topological space. And a priori, it seems possible to me that even a metric space whose metric is not spherically complete may still admit another compatible spherically complete metric.
We know that there exist spaces that have spherically complete metrics.
For example:
- All (complete) metrics on a compact space are spherically complete.
- The standard euclidean metric on $\mathbb{R}$ is spherically complete.
- All completely ultrametrizable spaces admit a compatible spherically complete metric.
I suspect that every locally compact completely metrizable space admits a spherically complete metric, and probably the same is true for Polish spaces (using that they are homeomorphic to $G_\delta$ subsets of the Hilbert cube). But I could not find any reference for this.
So to be more precise my question is:
- Do all completely metrizable topological spaces admit a compatible spherically complete metric?
- If not, is it known what is the largest class of (completely metrizable) topological spaces that admit one?
Even an answer/reference only for locally compact or Polish spaces would be nice. Thank you!
Edit (April 24th): If in a few more days there won't be an answer, I'll try cross-posting this on MathOverflow as well.