Definition of connectedness and it's intuition

If course any space $X$ having two or more points can be written as $A \cup B$, with $A,B$ disjoint and non-empty, in lots of ways. But being disconnected means that there is a way to do that such that no point of $A$ is "close to" $B$ and no point of $B$ is "close to" $A$. Being close to is formalised in topology by being in the closure. So call a space $X$ disconnected when we can write it as $A \cup B$, both sets non-empty and such that $\overline{A} \cap B = \emptyset$ (no point of $B$ is close to $A$) and $A \cap \overline{B} = \emptyset$ (no point of $A$ is close to $B$). But this implies that $$X\setminus B= A \subseteq \overline{A} \subseteq X\setminus B$$ so in particular $A=\overline{A}$ and $A$ is closed. Symmetrically, $B$ is closed too, and as $A$ and $B$ are each other's complements, $A$ and $B$ are open too (which you could also see as follows, e.g if $x \in A$ were not an interior point of $A$, every neighbourhood of $x$ would contain non-$A$ points, so points of $B$, as $A\cup B=X$. And if every neighbourhood of $x$ intersects $B$, $x \in \overline{B}$, but we assumed no point $x$ of $A$ was close to $B$...)

So we are at the definition of the question, calling a space that is not disconnected in this sense, "connected". It's in fact equivalent to ask in the disconnectedness definition for simultaneously open parts, simultaneously closed parts or "separated" parts (as the first definition).

If you cut some connected set into two pieces, then at the site of the cut, one of the two pieces will be "open", while the other will be "closed". For instance, if you cut the real line into two pieces at the point $a\in\mathbb R$, you will either get two pieces $(-\infty,a],(a,\infty)$, or $(-\infty,a),[a,\infty)$. At least one of them has a closed boundary at $a$. The points belonging to the cut need to be included in one of the two pieces, and that piece will have the cutting point as a boundary point. Similarly for more complicated spaces: the line along which we cut has to be distributed among the two pieces, giving them a boundary, making them not open.

Of course, we don't need to cut along a line/plane/whatever, but it's the case where the intuition is the most immediately clear.