## vrac sur la topologie issue des structures linéaires de Tim Maudlin

(via anniceris)
en attendant d’avoir le papier en question (celui-là si un hasardeux lecteur a l’accès académique adéquat… got it ! ), la vidéo de Tim Maudlin à propos de la topologie de structures linéaires qu’il propose comme fondation au concept d’espace temps relativiste (ce qui me reste à comprendre).
(à lire : time travel and modern physics par Frank Arntzenius et Tim Maudlin, Stanford Encyclopedia of Philosophy)

j’ai pompé quelques unes de ses slides ici pour essayer d’y réfléchir :

Axioms

A linear structure is set S together with \Lambda a set of subsets of S called the “lines” of S that satisfy
- LS1 minimality axiom : each “line” contains at least two points
- LS2 segment axiom : every “line” \lambda admits of a linear order among its points such that a subset of \lambda is itself a “line” if and only if it is an interval of that linear order
- LS3 point splicing axiom : if \lambda and \mu are “lines” that have in common only a single point p that is an endpoint of both, then \lambda \union \mu is a line provided that no lines inthe set (\lambda \union \mu) - p have a point in \lmabda and a point in \mu
- LS4 completion axiom : any linearly ordered set \sigma such that all and only the closed intervals in the order are “closed lines” is a line

Non uniqueness of order

according to the first set og axioms, every line can be represented by a linear order among its point. NBut evidently ther are two such linear orders that will do the job, one the inverse of the other. Each will imply the same intervals and do the same structure of “segments” ( a “segment of a “line” \lmabda is a subset of \lambda that is a “line”)

Neighborhoods
a set \Sigma is a “neighborhood” of a point p iff every “line” with p as an endpoint has a “segment” with p as an “endpoint” in \sigma

Open Sets
- a set \Sigma in a Linear Structure is an “open set” iff it is a “neighborhood” of all of its members.
(la différence avec la topologie standard, est qu’alors, le voisinage est défini comme un ouvert contenant le point : c’est ça en fait qui m’interpelle. ça semble en effet tellement plus intéressant d’adopter la démarche inverse)

Thm
The collection of “open sets” in a Linear Structure satisfies the aims of standard topology : the “open sets” are open sets !

–>> Directed Linear Structures
–> all and only directed intervals in a linear order are “segments” of a “line”etc..
-> splicing axiom ; final point and initial point

-> def outward neigborhood, outward open sets

—> de tout ça il déduit que c’est la topologie adaptée à l’espace-temps relativiste (lorentzian pseudo-metrique) maximal set of event forms such a set to intuitively form a line

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