On Tue, May 27, 2025 at 06:55:38PM +0100, Martin Baker wrote: > On 27/05/2025 16:31, Waldek Hebisch wrote: > > So, it seems that you want to represent topology via its lattice > > of open sets. > > Well, I'm keeping an open mind about open sets but my initial instinct was > to use: > * labeled open sets for topologies. > * set size + lattice for iso classes. > > The reason for the second is that, to use labeled open sets for iso classes, > the constructor algorithm would have to: > * choose a representative for each iso class (this seems very arbitrary and > I don't know how to do it) > * choose a basis. > * make sure these choices are closed under union and intersection.
There is no need to have basis closed under set union. Actually, it makes sense to have basis which is as small as possible, and on set of cardinality N it is possible to take basis which has at most N elements. Similarly, intersection of two members of basis does not need to be in basis, all what is needed is that intersection is an open set, that is can be written as sum of elements of basis. > (I think I should re-read your earlier messages in case you have explained > these things already) > > By the way, in you earlier email you mentioned using pre-order to code this, > this prompted me to read this page: > https://en.wikipedia.org/wiki/Specialization_(pre)order > So I am starting to understand what you meant. However it suggests that > Specialization pre order only works for T0 separation. No, pre-order works for arbitrary finite topological spaces. T_0 means that you get _partial order_. > At first I thought > this would not matter (why bother about points that can't be distinguished > by the lattice). Then I had the thought these points must be the degenerate > points. Things I've read suggest that degenerate points are very useful (for > example in calculating products of topologies and other nice properties) so > perhaps I do need T0. When looking at structure of topological spaces, spaces that are not T_0 are mostly a distraction, one can easily get properties of arbitrary topological spaces knowing structue of T_0 spaces. But since in arbitrary spaces two different points may be indistinguishable by topological properties alone it is less convenient to speak about such spaces. > I am thinking that I would need to base it on a weak > pre-order? Do you know if this speculation, on my part, is valid? I affraid that we use different terminology here. In the Wikipedia article (and I use the same definition) pre-order means that there may be two different points x and y such that x <= y and y <= x. That allows handling arbitrary topological spaces. Partial order means that when x <= y and y <= x, then x = y, which implies that spaces corresponding to partial orders are T_0. <snip example> > OK, I will have to work through this. It just seems counter-intuitive > because there are a lot less iso classes than topologies so we should be > able to code them with less data? Well, discarding information is not so easy. You need to invent some clever encoding, clearly lattice of open sets does not give you such an encoding. -- Waldek Hebisch -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/fricas-devel/aDYNWKWdrNqDJJkx%40fricas.org.
