On Thu, Feb 13, 2025 at 10:59:20AM +0000, Martin Baker wrote:
> The attached file contains my first tentative experiments with what I
> thought of as Finite Topology. Could you take a quick glance at it to
> see if that's what you would describe as Finite Topology, if not could
> you suggest an alternative name?
The 'OpenSet' domain is confusing/confused. First, you do not say
what is the meaning. Just saying 'open set' is almost meanigless,
set is open or not depending on topology. Second, looking
at imlemented functions it looks like Set(NonNegativeInteger),
with one or two added operations and missing several other
operations.
'TopologyFinite' looks like reasonable attempt to implement
topology on a finite set. But there are some warts. First,
implementation of 'intersection' and 'union' is clearly
bogus. Normal practice when you do not have reasonable
implemntation is to leave function unimplemented. Or
possibly write:
intersection(a:%, b:%) : % == error "intersection: unimplemented"
and similarly for all other functions without good implementation.
Second, 'setTopologyFromList' "on faith" treats given list as
a correct topology. Normal practice in FriCAS is either to
check that this is really a topology or generate topology
from given list (assiming that this is list of open sets,
we can take intersections and than sums of intersections to
obtain topology). Third, {{}, {1}, {1, 12}, {1, 20}, {1, 12, 20}}
gives a topology on set {1, 12, 20}, but your code seem to
assume that the space is really interval from 0 to 20.
It is not clear to me what TopoSpaceFinite is supposed to mean?
Do you want points of your space to be symbols? If yes, you
can just take type of points as parameter of constructor, no
need to repeat definition. If you want something else, then what
really want?
> Anyway its the subject that interests me at the moment and I think it
> should keep me busy for a few years.
>
> Despite what you say about the usefulness of 'geometric realization' I
> would still like to attempt to implement it, just for completeness and
> my own education. I realise that n points will produce 2*n+1 dimensions
> but FriCAS can handle high dimensional matrix algebra, right? That's one
> of the advantages of doing this on a CAS.
You can do something, but since there is no embedding for general
finite topology, you can not implement embedding. To make it a
bit stronger, consider topolony on {0, 1} where open sets are
{{}, {1}, {0, 1}}. Every continuous function from this space
to euklidean space is constant. You probably can "unnerve" your
space, that is find collection C of open sets such nerve of C
is the same as your space. But that is different than
embedding and in mathematics such differences matter quite a lot.
--
Waldek Hebisch
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