On Wed, Feb 12, 2025 at 4:31 PM Waldek Hebisch <de...@fricas.org> wrote:
>
> On Wed, Feb 12, 2025 at 04:06:35PM -0600, Dima Pasechnik wrote:
> > On Wed, Feb 12, 2025 at 12:52 PM Waldek Hebisch <de...@fricas.org> wrote:
> > >
> > > On Tue, Feb 11, 2025 at 02:44:20PM +0000, Martin Baker wrote:
> > > > On 09/02/2025 17:45, Waldek Hebisch wrote:
> > > > > Finite topological spaces are equivalent to partial orders, so
> > > > > there is connection to logic. But they are quite different than
> > > > > infinite topological spaces like simplicial complexes.
> > > >
> > > > I hope there is a way to implement a 'geometric realization' function,
> > > > that is an algorithm to embed a finite topological space in a vector
> > > > space (which is a Euclidean space, which is a topological space).
> > >
> > > I am affraid there is confusion what "finite topological space"
> > > means. By this I mean finite set with topology. Simplest
> > > nontrivial example is two element set {a, b}, with topology
> > > {{}, {a}, {a, b}}. Nontrival here means that this in neither
> > > discrete topology nor anti-discrete one. There is a theorem
> > > saying that any finite subset of euclidean space has discrete
> > > topology. So, to have non-trivial topology on a subset of
> > > euclidean space you must have an infinite set. So
> > > 'geometric realization' can only work for "nice" spaces
> > > and gives interesting results only for infinite ones.
> > > Actually, there is notion of topological dimension which
> > > for separable metric spaces say that space of dimension n
> > > can be topologically embedded in euclidean space of
> > > dimension 2*n + 1.
> > >
> > > Note that "finite simplicial complexes" are typicall
> > > infinite topological spaces, the word "finite" means
> > > that there is finite number of pieces, but typicall
> > > some pieces are infinite.
> >
> > the basic notion is an "abstract" simplicial complex.
> > It should not be confused with a "geometric simplicial complex" (an
> > embedding of an abstract one into a space of some sort, e.g an
> > Euclidean space.)
> >
> > A finite abstract simplicial complex is a purely combinatorial object.
> > One can study its homology groups, over a finite field (e.g.
> > Z_2-homology is very common)
> > without resorting to Euclidean spaces.
> > E.g. SageMath can compute such things:
> >
> > S = SimplicialComplex([[0,1], [1,2], [0,2]]) # circle
> > T = S.product(S) # torus
> > Simplicial complex with 9 vertices and 18 facets
> > sage: T.homology(base_ring=GF(2))
> > {0: Vector space of dimension 0 over Finite Field of size 2,
> > 1: Vector space of dimension 2 over Finite Field of size 2,
> > 2: Vector space of dimension 1 over Finite Field of size 2}
>
> Sure. Martin was writing about finite topological spaces.
> I hope you are not considering the finite complex above as
> a topological space.
Sure, an abstract finite simplicial complex A is a combinatorial model
of a finite triangulation of whatever space S we're studying,
in other words, of a geometric simplicial complex homeomorphic (or
perhaps only homotopy equivalent) to S.
A is not a finite topological space by itself (finite topological
spaces are kind of boring).
My point was that often one can extract a lot of info from A, without
dealing with S itself.
Dima
>
> --
> Waldek Hebisch
>
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