Topology, projection map

1. Nov 11, 2004

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?

2. Nov 11, 2004

Hurkyl

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

3. Nov 11, 2004

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.

4. Nov 12, 2004

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.