Example of two closed continuous functions whose "product" is not closed

Question

Let $X,Y,W,Z$ be topological spaces, and let $f: X \longrightarrow Y$, $g: W \longrightarrow Z$ be closed functions. Let $f \times g: X \times W \longrightarrow Y \times Z$ be such that $f \times g(x,w)= (f(x),g(w))$.

1) Find $X, Y, Z, W$, $f$ and $g$ (closed) such that $f \times g$ is not closed.

2) Find $X, Y, Z, W$, continuous (and closed) functions $f$ and $g$ such that $f \times g$ is not closed.

Edit: I was able to solve the problem when I tried to formalize that an example with $\mathbb{R}$ with the euclidean topology couldn't work. I believed that $f:\mathbb{R}\longrightarrow \mathbb{R}$ is closed and continuous iff its limit for $x \longrightarrow \pm \infty$ is $\pm \infty$, hence forgetting the case were $f$ is definitively costant, which is useful to build the desired example.

And the task must be to find $X,Y,Z,W$ together with $f$ and $g$, rather than to construct $f$ and $g$ when $X,Y,Z,W$ are given. For example if they are all one-point spaces, then the task is clearly impossible! — hmakholm left over Monica, Dec 27 '14 at 19:12
This question is not quite the same, but the answer may give some useful ideas nevertheless. — hmakholm left over Monica, Dec 27 '14 at 19:35

score 2 · Accepted Answer · answered Dec 27 '14 at 21:46

Take $X, Y, W, Z = \mathbb{R}$ with the euclidean topology;

$f(x)\equiv 0$ and $g(x)=x^2$ are obviously continuous and it is easy to check they are closed. Moreover, $f \times g$ maps the closed set $\lbrace (n,\frac{1}{n})\vert n \in \mathbb{N} \rbrace$ to the set $\lbrace (0,\frac{1}{n^2})\vert n \in \mathbb{N} \rbrace$, which is not closed, as desired.

Example of two closed continuous functions whose "product" is not closed

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