While reading some books, I often find the following argument:
Consider $A\subset X$, when $X$ is a topological space.
If $A$ is a disconnected set, then there exist some open sets, say $U,V \in \tau_X$ disjoint and non empty such that $A = U \cup V$.
But this often confuses me, because the part when it says "open sets of $\tau_X$" because, for example, $[0,1] \cup [5,6]$ is disconnected and it can't be written as union of two open sets -of X!- which are disjoint and non empty.
The definition in some books of disconnected set $A$ says that $A$ is disconnected if it is disconnected with the subspace topology.
But then, I don't understand why some authors take open sets of the big space, say $X$.
Can someone help me to clarify this?