An image suitable for a skyscraper sheaf?

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:

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.

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.
• 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 $\{g\}$. Nov 21 '15 at 11:20