The Universe of Discourse

Yesterday's metric has a name

Yesterday I presented as a counterexample the topology induced by the following metric:

I asked:

It seems like this example could be useful in other circumstances too. Does it have a name?

Several Gentle Readers have written in to tell me that that this metric is variously known as the British Rail metric, French Metro metric, or SNCF metric. (SNCF = Société nationale des chemins de fer français, the French national railway company). In all cases the conceit is the same (up to isomorphism): to travel to a destination on another railway line one must change trains in London / Paris, where all the lines converge.

Wikipedia claims this is called the post office metric, again I suppose because all the mail comes to the central post office for sorting. I have not seen it called the FedEx metric, but it could have been, with the center of the disc in Memphis.

[ Addendum 20180621: Thanks for Brent Yorgey for correcting my claim that the FedEx super hub is in Nashville. It is in Memphis ]

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Bounded does not imply totally bounded

I somehow managed to miss the notion of totally bounded when I was learning topology, and it popped up on stack exchange recently. It is a stronger version of boundedness for metric spaces: a space !!M!! is totally bounded if, for any chosen !!\epsilon!!, !!M!! can be covered by a finite family of balls of radius !!\epsilon!!.

This is a strictly stronger property than ordinary boundedness, so the question immediately comes up: what is an example of a space that is bounded but not totally bounded. Many examples are well-known. For example, the infinite-dimensional unit ball is bounded but not totally bounded. But I didn't think of this right away.

Instead I thought of the following rather odd example: Let !!S!! be the closed unit disc and assign each point a polar coordinate !!\langle r,\theta\rangle!! as usual. Now consider the following metric:

$$ d(\langle r_1, \theta_1\rangle, \langle r_2, \theta_2\rangle) = \begin{cases} r_1, & \qquad \text{ if $r_2 = 0$} \\ \lvert r_1 - r_2 \rvert, & \qquad\text{ if $\theta_1 = \theta_2$} \\ r_1 + r_2 & \qquad\text{ otherwise} \\ \end{cases} $$

The idea is this: you can travel between points only along the radii of the disc. To get from !!p_1!! to !!p_2!! that are on different radii, you must go through the origin:

Now clearly when !!\epsilon < \frac12!!, the !!\epsilon!!-ball that covers each point point !!\left\langle 1, \theta\right\rangle!! lies entirely within one of the radii, and so an uncountable number of such balls are required to cover the disc.

It seems like this example could be useful in other circumstances too. Does it have a name?

[ Addendum 2018-07-18: Several Gentle Readers have informed me that this metric has not just one name, but several. ]

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