I'm reading through wikipedia for a rigorous definition of disconnected topological space, which is the same as the one given by Munkres.
A topological space $X$ is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, $X$ is said to be connected.
The question is: does the bit "if it is the union of two disjoint non empty open sets" mean that the family of open sets defining the topology on $X$ cannot be partitioned into two disjoint non empty open set? Or maybe more specifically that if I take an open set $\cal{O}_1$ and an open set $\cal{O}_2$ the union of these is not $X$?
Let $X$ be a set with topology $\tau$.
$X$ is disconnected if there exist two disjoint non-empty open sets (elements of $\tau$) whose union is $\pmb X$.
This is not about partitioning $\tau.$