Is a disconnected topological space not connected? (Elementary) This may be obvious (or not), but I am trying to prove that something is not connected, but then I remembered that a topology that is 'not open' does not mean that it is 'closed' and that perhaps I should be a little careful before assuming that I can treat 'not connected' as 'disconnected'.
Kind regards!
 A: In this case there is no need to careful: connected means "not disconnected" by definition. So logic dictates that "disconnected" is the same as "not connected".
Recall that we define a disconnected space first, in most texts that I know, namely $X$ has a partition $(A,B)$: $X = A \cup B$ where $\emptyset \neq A, \emptyset \neq B, A \cap B = \emptyset$ and both $A$ and $B$ are open (or equivalently, closed). Then $X$ is defined to be connected if no such partition exists, otherwise (if one exists) it is disconnected. A subset $A \subset X$ is called connected if it is connected in its subspace topology.
More "positive" reformulations of connectedness exist, e.g. (TFAE):


*

*$X$ is connected.

*Every continuous function from $X$ to $\{0,1\}$ (in the discrete topology), is constant.  

*For every open cover $\mathcal{U}$ of $X$, there exists a chain between any two points of $X$: $$\forall x,y \in X: \exists U_0,\ldots, U_n \in \mathcal{U}: (x \in U_0 \land y \in U_n \land \forall i \in \{0,\dots,n-1\} U_i \cap U_{i+1} \neq \emptyset)$$ 

