Why did Munkres write "unless one of the spaces $X$ or $Y$ happens to be empty"? On p.87 in "Topology 2nd Edition". I am reading "Topology 2nd Edition" by James R. Munkres.
On p.87, there is the following definition:

Definition. Let $\pi_1 : X \times Y \to X$ be defined by the equation $$\pi_1(x, y) = x;$$ let $\pi_2 : X \times Y \to Y$ be defined by the equation $$\pi_2(x,y)=y.$$
The maps $\pi_1$ and $\pi_2$ are called the projections of $X \times Y$ onto its first and second factors, respectively.
We use the word "onto" because $\pi_1$ and $\pi_2$ are surjective (unless one of the spaces $X$ or $Y$ happens to be empty, in which case $X \times Y$ is empty and our whole discussion is empty as well!).

For example, if $X = \emptyset$, then $X \times Y = \emptyset$.
So if $Y \ne \emptyset$, then $\pi_2$ is the empty mapping $\emptyset \to Y$.
So $\pi_2$ is not surjective.
But,
for example, if $X = \emptyset$, then $X \times Y = \emptyset$.
So $\pi_1$ is the empty mapping $\emptyset \to \emptyset$.
So $\pi_1$ is bijective.
So $\pi_1$ is surjective.
Why did Munkres write "unless one of the spaces $X$ or $Y$ happens to be empty"?
 A: If $X\ne\emptyset$ and $Y=\emptyset$, then $X\times Y=\emptyset$ and $\pi_1$ is the empty map, which is not surjective. Symmetrically if $Y\ne\emptyset$ and $X=\emptyset$.
There might be a nuance in the English interpretation of "unless", though I am certain Munkres, nor any English-speaking mathematician, would use "$A$ unless $B$" as synonymous with "$A$ if and only if not $B$". Therefore let's say that, if you want as precise a statement as it can be, it should be that $\pi_1$ and $\pi_2$ are both surjective if and only if either both spaces are non-empty or they both are empty. Or, even better, that the projection to one of the factors is surjective if and only if that factor is empty or the other one isn't.
A: I don't see any problem. The statement "$\pi_1\text{ and }\pi_2\text{ are surjective}$" is not true when either $X\text{ or }Y$ is empty, since then atleast one of $\pi_1,\pi_2$ are not surjective as you have seen. Note that the negation of "$\pi_1\text{ and }\pi_2\text{ are surjective}$" is "$\pi_1\text{ OR }\pi_2\text{ is not surjective}$".
