This question relates to this thread:

Skyscraper sheaf?

Consider one of the diagramms for the representation of a sheaf (and stalks thereof) which are popular on the web:

enter image description here

I just wanted to know whether the representation on each point represents a closed or an open point and whether it is in fact the representation of a skyscraper sheaf.

Thanks in adavance.


This does not depict a skyscraper sheaf. As explained in the description here, it represents a sheaf over a discrete space $X=\{r,g\}$, i.e. where both points are open and closed at the same time. A section over the open set $\{r\}$ corresponds to a choice of a red point, and a section over the open set $\{g\}$ corresponds to a choice of a green point.

  • $\begingroup$ Thanks for your reply. I have an important doubt, though, namely, how can a point be open and closed at the same time? The only thing I can think of is if it is an empty set. If there is more to it than that, please explain..... $\endgroup$ Nov 21 '15 at 11:09
  • $\begingroup$ The discrete topology on a set is defined by declaring every subset to be open. You can check that it is indeed a topology. But then any set is also closed, its complement being open. In your case, we can say that both $\{r\}$ and $\{g\}$ are open sets by definition, and as a consequence we have that e.g. $\{r\}$ is closed being the complement of $\{g\}$. $\endgroup$
    – Andrea
    Nov 21 '15 at 11:20
  • $\begingroup$ I guess my question is: would you say each of the green or red dots in the stalk are open and closed at the same time? $\endgroup$ Nov 21 '15 at 11:24
  • $\begingroup$ Those are not points of your topological space, but something attached to it, so the question does not make sense (unless you consider them as points in the étalé space of the sheaf, but that is another story). The topological space, in the picture, is just the blue disk, and its two points are the black dots. $\endgroup$
    – Andrea
    Nov 21 '15 at 11:30
  • $\begingroup$ Well, I am interested in the procedures of selecting some point from the red stalk and another from the green stalk and then gluing them together. I guess for them to be selected the stalk must be a set of some type, right? Is it an open set of open points? A closed set of closed points, or what exactly? $\endgroup$ Nov 21 '15 at 11:36

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