Connectedness. Definition

Connectedness, motivation

Focus on two sets that union to create the set in question: if two sets union to the whole but are not "touching", then the set in question is disconnected. Intuitively, a space is said to be 'connected' if it is all in one piece—it does not fall naturally into two or more pieces. 'Naturally' here carries a major component of the sense of 'connected', as any topological space can be torn into multiple pieces in some way.

Disconnected

The idea of a set being disconnected is possibly more intuitive than the idea of a set being connected. In $\mathbb{R}$, a set $E\subseteq \mathbb{R}$ is disconnected if it can be expressed as the union of two separated sets. Separated here means that the sets share no limit points ($A$ and $B$ are separated if $\bar{A}\cap B$ and $A\cap \bar{B}$ are empty). In other words, one can't find a sequence in $A$ whose limit is in $B$ and vise versa.

Yet another formulation is to say that $E\subseteq \mathbb{R}$ is disconnected iff for some $a,b\in E$ there is a $c\in \mathbb{R}$ but not in $E$ such that $a<c<b$. Push a negation into this statement to get back to the connected version: $E$ is connected iff for all $a,b\in E$, whenever $a<c<b$ for some $c\in \mathbb{R}$, then $c\in E$. 

Tearing

Intuitively, why can't the real line $\mathbb{R}$ be separated naturally into two pieces? Consider the decomposition $(-\mathrm{\infty },0)\cup [0,\mathrm{\infty })$: in what way is this separation not natural? While it might not appeal to everyone's intuition, this separation is considered forced because a limit point of $(-\mathrm{\infty },0)$, zero, is not an element of $(-\mathrm{\infty },0)$ but an element of the other piece $[0,\mathrm{\infty })$. In other words, the closure of one piece spills into or is contained or is intertwined in the other piece. Thus, naturally separating a space can be viewed from the lens of open and closed sets.

Natural separation, requirements

For a topological space $X$, a separation of $X$ into disjoint subsets is considered a natural separation into independent pieces $U$ and $V$ if no point in $U$ is a limit point of $V$, and vice versa. Equivalently, this means that $Cl(U)$ is disjoin from $V$, and vice versa, or equivalently, that both $U$ and $V$ are closed, or equivalently yet again, that both $U$ and $V$ are open.

Thus, we arrive at our definition of connectedness, by defining a space $X$ connected if the only natural separation of $X$ as described above is the trivial one, $\mathrm{\varnothing }$ and $X$ itself.

Sufficiency of open sets

If disjoint open sets are sufficient to cover a space, then the space is not connected. Open sets don't contain their boundary limits, and two disjoint open sets must therefore not be touching. There is at least a point that exists in the sense of limits that is in neither of the open sets. In this sense, the space is disconnected.

Source