Hi Waldek,
On 03/05/2025 13:17, Waldek Hebisch wrote:
Number of topologies grows very fast with n, any method which tries
to generate all topologies and count them is going to be inpractical
even for moderately large n.
I could easily create a version of the code that does not generate a
list (either just outputting a count or print to a stream or something
like that). That would save a lot of memory but the recursive code would
still store a representation of a big part of the lattice structure in
memory at any one time. There might be a way to generate subsets and
then put them all together.
I probably don't want to spend much time optimising the SPAD code, I'm
more interested in the underlying algorithm. If the algorithm is correct
perhaps one day computers will improve enough to run it with bigger sets.
Any ideas how I could check this out further?
1) Use more compact representation of topology than list of sets.
2) Count only T_0 topologies, count only representatives of
homomorphism classes
I have started to think about how the homomorphism classes could be
generated by decomposing unlabeled sets instead of the current code
which decomposes labeled sets.
I have put my initial thoughts here:
https://www.euclideanspace.com/maths/topology/topologicalSpace/finite/inequivalent/index.htm
This is leading me toward the idea of representing as a partial order
which links to what you said.
Most topologies are likely to have small stabilizers, so the
formulas above are likely to require much less work than
generating all topologies.
Yes, I have an intuition there is some parallel here to the way that
permutation groups are represented in FriCAS/Axiom/Scratchpad. Also I
haven't fully thought through how this links to homotopy groups.
Thanks,
Martin
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