How many open covers does the empty topological space have? Not one, not none, but two: the empty cover $\varnothing$, since its union is $\bigcup\varnothing=\varnothing$, and the cover {$\varnothing$}, since its union is also $\bigcup${$\varnothing$}$\ =\varnothing$.
This comes up when using the Grothendieck plus-construction to sheafify a presheaf. Apply the construction to the (nonseparated) presheaf $P:\mathcal{O}(X)^{op}\to \mathrm{Set}$ sending every open set to the set $A$, with $|A|\geq 2$. Then the presheaf $P^+:\mathcal{O}(X)^{op}\to\mathrm{Set}$ agrees with $P$ on every open set except $\varnothing\subseteq X$, where $P^+(\varnothing)$ is now a one-element set {$*$}. This is because the matching families for the cover {$\varnothing$} of $\varnothing$ (of which there is one for each $a\in A$) are all set equal to the unique matching family for the refining cover $\varnothing\subseteq\ ${$\varnothing$} of $\varnothing$.
This elementary example comes from "Sheaves in Geometry and Logic", by Moerdijk and MacLane.
