Proposition 5

For a topological space $X$ to be not connected it is necessary and sufficient that there exists a surjective continuous mapping of $X$ onto a discrete space containing more than one point.

The condition is sufficient by Proposition 4. Conversely, if $X$ is not connected, there exist two non-empty disjoint open subsets $A, B$ whose union is $X$, and the mapping $f$ of $X$ onto a discrete space of two elements $\{a, b\}$, defined by $f(A) = \{a\}$ and $f(B) = \{b\}$, is continuous.