If C is a closed subset of the set $X \times Y$ then is $\pi_{1}(C)$ closed also?

Question

Let $X$ and $Y$ be topological spaces. Let $X \times Y$ be given the product topology, let $\pi_{1}:X\times Y \to X$ be the canonical projection.

Now suppose $C$ is a closed subset of $X \times Y$, does $\pi_{1}(C)$ necessarily have to be closed?

I have been told you have to use the fact that the projection of open rectangles is open in $X$. As well as the fact that the projection of open subsets of $X \times Y$ are also open in $X$, but I can't figure it out still.

score 4 · Answer 1 · answered Mar 08 '18 at 19:15

4

No, a standard example is $X=Y=\Bbb R$ and $C$ the graph of $xy=1$.

answered Mar 08 '18 at 19:15

Angina Seng

158,341

How can we set $C$ as $xy = 1$? Can we treat this function as a closed set in the product topology? – user501050 Mar 08 '18 at 20:02
1

@user501050 Recall I said "the graph of ..." – Angina Seng Mar 08 '18 at 20:04

score 0 · Answer 2 · answered Mar 08 '18 at 19:35

0

This will hold for all $C$ (i.e. $\pi$ is a closed map) iff $Y$ is compact. I show this in this answer, e.g.

answered Mar 08 '18 at 19:35

Henno Brandsma

242,131

If C is a closed subset of the set $X \times Y$ then is $\pi_{1}(C)$ closed also?

2 Answers2