> My take is that the two entities don't adjoin each other at all,
> since the length of their border is zero

As you might guess from my cheery request for other people's
opinions, I "posted" this almost a day ago, before so many others had
their say, but it's only now appeared.

I still think a border of zero length isn't a border in any
conventional way at all. But life gets tricky around infinitesimals.
I think Bill's comments

> Michael, I suppose this may be over-simplification, but
> in "traversing" the point singularity, one would leave behind one
> entity for another...therefore, the point would be a border by
> definition, regardless of its topology.

are very interesting, though, and they precisely address my
difficulty with borders containing quadripoints. Surely the various
tri-, quadri- and higher points represent discontinuities between
borders, in the same way that borders are discontinuities between
countries? Move across a border, and you leave one country and enter
another. Move *along* a border, cross a tripoint, and you're
following a different border (choice of two, in this case).
So 0-dimensional points form the boundaries between different 1-
dimensional borders, which form the boundaries between different 2-D
countries.

Grant