Doubt about Gluing Topological Spaces in Vakil’s FOAG

Howdy, pals. 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.

The issue is: he says that:

$ U sim V$

As the equivalence relation, but without defining it. I think that the equivalence is via the homeomorphism, hence the problem is to understand the classes, I was guessing something like:

$ left { V bigcup varphi^{-1}(V) right }$

Where $varphi$ is the homeomorphism from U to V. I just need some help to understand this equivalence relation and the classes.

submitted by /u/magikarpa1
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Nevin Manimala

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