What are the properties of open sets in X x Y for a continuous projection map?

[sparkster]

I'm trying to prove some stuff that involves the projection map, say p:X x Y ->X. But I need to know if it's continuous. If a map is continuous, then the preimage of a open/closed set is open/closed.

The problem is, what do open sets in X x Y look like? I know what the basis elements are, and the open sets would be arbitrary unions and finite intersections, but is there any way to generalize?

 

[Hurkyl]

You don't need to know all open sets, you just need to show that some particular sets are open.

 

[sparkster]

Well, yeah, but I thought that if there was a way to list all the open sets, that would take care of it.

Ok, if I take p(A) to be open, I need to show that the preimage is open. The preimage would be A x B, where A is open in X. If B is open, then I'm done. But if B is closed, I don't know what to do.

 

[cogito²]

So p: X \times Y \to X and p((x,y)) = x. But now think about it what sets will give you \{x\} as an image? p((x,y)) = x for all y \in Y. So p^{-1}(\{x\}) = \{x\} \times Y. I think you can take it from there.