y'all. I'm doing (trying to) FOAG on my own and in chapter 4 when Vakil do (actually, asks to) gluing schemes, he first defines how to glue topological spaces, but his definition is very different from any other books. I know that I can do the exercise 4.4A without understand his definition on how to glue topological spaces, but since I'm studying alone the notes I was not wanting to move forward without truly understand it. The definition is here (http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf) on page 141.
I'm having some trouble to understand the classes. Usually the books define the equivalence relation on points. So if anyone could shed some light I would be grateful. Just to be clear: I'm trying to understand the classes on Vakil's definition.
Best Answer
The new topological space is $W$, whose points are equivalence classes of points from $X \cup Y$, each of which is either a singleton (if it is a point of $X$ not in $U$, or a point of $Y$ not in $V$) or a pair of points (a point of $U$ and the corresponding point of $V$).
The open sets of $W$ are those sets of the form an open set from $X$ union an open set from $Y$.