On 2018-06-03 19:22, Vaughan Pratt wrote:
...
An affine space over any given field differs *k* from a vector space over *k* only in its algebraic structure, not its topological structure.
In these discussions, the "relation and interplay" between algebra and topology, and with logic for that matter, again becomes almost a matter of choice. There is the traditional algebraic view "of algebraic operations", which is narrower with respect to including "algebraic operations on sets". This again comes to category theory and "underlying categories". Pratt's contributions to this mailing list are always clever and very demanding e.g. to me confessing I seldom understand even fractions of the subtleties (... and other things in his previous postings) Vaughan is presenting and writing, so to hide my shortcomings I might see it so that we could even complicate things, as we may need not just to distinguish the topological structure of an algebra from the original algebraic structure, but also consider the algebraic structure of that topological structure of that algebraic structure. Those two algebras are not the same. When everything is over Set (the category of sets), we do not possess the machinery to unravel this, is my credo, but when working over appropriate categories, other and expanded views open up. They may or may not be useful, and that is often a topic for "applications", real-world or otherwise unreal. Order, and non-commutativity, obviously, also comes into play. --- ADVERTISEMENT (if I may): https://www.springer.com/us/book/9783319789477 --- Thanks, Vaughan. Cheers, Patrik [For admin and other information see: http://www.mta.ca/~cat-dist/ ]