Showing that projections $\mathbb{R}^2 \to \mathbb{R}$ are not closed

Consider $\mathbb{R}^2$ as $\mathbb{R} \times \mathbb{R}$ with the product topology. I am simply trying to show that the two projections $p_1$ and $p_2$ onto the first and second factor space respectively are not closed mappings. It seems like this should be easy, but I have not been able to come up with a closed set in $\mathbb{R}^2$ whose projection onto one of the axes is not closed.

I don't really have any work to show...I've really just tried the obvious things like closed rectangles and unions of such, the complement of an open rectangle or union of open rectangles, horizontal and vertical lines, unions of singletons, etc., and haven't come up with anything non-obvious, which I hope is where the answer lies. It's bothering me that I can't come up with an answer, and I'd appreciate some help. Thanks.

Solution 1:

No bounded set will work, because a closed and bounded set in ${\bf R}^2$ is compact, and the image of a compact set under any continuous map is compact (so closed in any Hausdorff space, in particular in $\bf R$).

On the other hand, the graph of any function with a vertical asymptote will work, for instance that of $1/x$.

In fact, it is not hard to show that any open set in $\bf R$ can be obtained as the projection of a closed set in ${\bf R}^2$ in such a way that the projection is injective (no two points in the closed set project onto the same point in $\bf R$), by a similar technique. This is related to the classical fact that any $G_\delta$ (countable intersection of open sets) in ${\bf R}$ (or any other Polish space, that is, separable and completely metrizable, if you're familiar with the concepts) can be embedded as a closed set in $\bf R^N$ (product of countably infinitely many copies of $\bf R$).

Solution 2:

The graph of $\tan^{-1}(x)$ is closed, but its projection onto the $y$-axis is not.

There's a related discussion in SO at Projection map being a closed map.

Solution 3:

Consider the map $\phi : \mathbb{R}^2 \to \mathbb{R}$ that takes: $$(x,y) \mapsto x \cdot y.$$

This map is continous, thus $\phi^{-1}(1)$ is closed in $\mathbb{R}^2$. On the other hand its projection onto the first coordinate is $\mathbb{R} -\{0\}$ which is not closed because it's open and $\mathbb{R}$ is connected.