Disconnected (topology)/Definition

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Mendelson and Lee's topological manifolds have it covered, I think Munkres is where I got "separation" from

Definition

A topological space, $(X,\mathcal{ J })$, is said to be disconnected if:

• $\exists U,V\in\mathcal{J}[U\ne\emptyset\wedge V\neq\emptyset\wedge V\cap U=\emptyset\wedge U\cup V=X]$, in words "if there exists a pair of disjoint and non-empty open sets, $U$ and $V$, such that their union is $X$"

In this case, $U$ and $V$ are said to disconnect $X$ and are sometimes called a separation of $X$.