On 26/05/2025 21:31, Waldek Hebisch wrote:
* isomorphism classes are sets of _unlabeled_ sets
(both with the usual meet,join top and bottom axioms)
I do not think this will work well (at least not in a simple
way). In isomorphism class you allow arbitrary relabeling, so
at "absolute" level labels do not matter. But they
matter in relative sense. You need to know when two elements
are in fact one element (that is they are equal), you need
to know if for given p and q there is an open set containing
q but such that p is not in this set. Easiest way to
keep this information is to use labels. You could try to
encode elements of the set by their properties, but to
do this you need to know which combination of properties
are possible.
What you wrote above looks more like you would like to represent
lattice of open sets as an abstract lattice. AFAICS you still
would have to decide when two abstrace lattices are isomorphic
and it is not clear if isomorphism class of lattices gives you
isomorphism class of topologial spaces. Mathematical version of
Murphy principle says that if there is no reason for some property
to be true, then it is false. And ATM I see no reason to
have nice correspondence there (of course I may be missing
something).
Perhaps if I use an example:
A topology might be represented like this:
[{}, {3}, {1, 2}, {1, 2, 3}]
But for a isomorphism class we want a representation that includes
multiple topologies such as:
[{}, {3}, {1, 2}, {1, 2, 3}]
[{}, {1}, {2, 3}, {1, 2, 3}]
and so on
We could choose one to represent the whole class:
[{}, {3}, {1, 2}, {1, 2, 3}]
Alternatively we could represent it as a partial order:
[ _|_<a , _|_ < b , a<T, b<T]
Or we could represent it as a lattice:
[ a/\b=_|_ , a\/b=T]
It seems to me that all 3 of these represent the same structure (upto
isomorphism).
I have used labels for the partial order and lattice cases but these
labels represent elements of the topology not elements of the open sets.
(So, in this case _|_={} a={3} b={1, 2} T={1, 2, 3})
So I think the partial order seems more compact and involves less
arbitrary choices.
Well, sometimes discarding information is not easy. Concerning meets
and joins, correspondence that I have in mind is 1-1 correspondence
between topologies and pre-orders. Any pre-order has corresponding
topology, so there is no requrement for nice meets and joins.
I don't quite see what you are saying here? Cant we define it either way
and convert between them:
[ _|_<a , _|_ < b , a<T, b<T] = [ a/\b=_|_ , a\/b=T]
I seem to remember that things get a bit more complicated for infinite
topologies but is what you say true for finite topologies?
It is not clear to me what _exactly_ you want to do, but it looks
tricky to make it work without using a representative of
something.
I would like to implement two domains, one for the topology case and
another one for the iso class case.
For the iso class case I cant see why the alternative codings I
suggested above would not work? They don't seem to require an arbitrary
representative.
I have been thinking for some time now that the most efficient way to
represent a topology would be to build it up from sub-topologies but I could
not work out how to do this. However, I have just realised how to do it:
build a big topology from smaller _discrete_ topologies.
So this:
[{1, 2, 3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {1, 2},
{1, 3}, {2, 3}, {1, 4}, {1}, {2}, {3}, {}]
could be coded like this:
P{1, 2, 3}+{1, 4}+{1, 2, 4}+{1, 3, 4}+{1, 2, 3, 4}
Where P means powerset.
Well, there is easy thing to do: each finite topological space
is a disjoint sum of connected componests. I the case above
one connected component is {1, 4} (with topology [{}, {1}, {1, 4}]),
there two other components: {2} and {3}. You can view {1, 4}
with the topology above as an extention of {1} by {4}, but in
general added part ({4} here) is "glued" to previous part in
a non trivial way. I think it is much clearer when you look
at corresponding pre-orders. Pre-order on {1, 4} is just
normal linear order, but it should be clear that you may have
much more "interesting" pre-orders. Let X be a topological
space. The set of minimal elements of X, call it M is just a
discrete space. X - M gives you smaller topology. But whole
topology is only determined once you now order relations between
elements of M and X - M.
This seems to me potentially a very efficient encoding for the topology
case. I will think about this more.
Martin
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