In the definition of the stalk of a presheaf of abelian groups on a topological space, at a point, one uses the fact that the open sets containing that point form a poset, which is directed via reverse inclusion. My question is, that set is directed via ordinary inclusion as well, right? So why use the reverse ordering instead of the ordinary one, to define the direct limit? Thanks!

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    $\begingroup$ If your index set has a maximal index, the direct limit is simply the object associated to that maximal index. So you would simply get the sections associated to the whole space. $\endgroup$ – Richard D. James Feb 24 '15 at 15:12
  • $\begingroup$ oh of course! Thanks!! $\endgroup$ – juremanet12 Feb 24 '15 at 15:15
  • $\begingroup$ Ah, I guess you mean maximal element, right? @SpamIAm $\endgroup$ – juremanet12 Feb 24 '15 at 15:28

Directed sets are called that because they have a direction, and for taking stalks, the direction you want to go is closer and closer to the point, not farther and farther away (which is what you get with ordinary inclusion).

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