Empty space connected? If so, then what are its components? I encountered that a topological space $X$ is connected if no separation exists. Here
a separation is a pair of disjoint non-empty open sets whose union
is $X$. Such a separation can only exist if $X$ contains two distinct
elements, so $\emptyset$ is supposed to be connected (right?). But
what about its components? Does it have $\emptyset$ as unique component
or are there in this case no components at all? They should form a
partition of $\emptyset$, and I was taught that elements of a partition
are non-empty.

Firstly, let us suppose that the space $X$ is non-empty. The condition of being connected then becomes equivalent to any of the following:
(a) if $X = Y \coprod Z$, then either $Y$ or $Z$ is canonically isomorphic to $X$,
(b) if $X \xrightarrow{f} (Y \coprod Z)$ then $f$ factors through one of the coproduct injections,
(c) the hom-functor $\mathbf{Top} \xrightarrow{\mathrm{hom}(X, -)} \mathbf{Set}$ preserve coproducts.
The statements are nice and should then be taken as a definition. However, when the empty space $\emptyset$ is taken into account one needs to add to the statements (a) the additional clause $X \neq \emptyset$ to make the equivalence work.
Thus, the empty space $\emptyset$ should  not be connected!
