connected space
A space is connected if it is all one piece — you cannot cleave it into two separate worlds with a clean gap between them. The real line is connected; two disjoint intervals laid side by side, with daylight in between, are not. Connectedness formalizes the intuition of a space having no internal break.
The precise definition is stated by what is forbidden. A space X is connected if it cannot be written as the union of two disjoint nonempty open sets. Equivalently, the only subsets of X that are simultaneously open and closed (the clopen sets) are the empty set and X itself; any other clopen set would split the space.
Connectedness is a topological invariant — preserved by continuous maps and by homeomorphisms — and it underwrites the intermediate value theorem: a continuous real function on a connected domain takes every value between any two of its values, precisely because its image, being the continuous image of a connected set, is itself connected and therefore an interval. A subtle point: connected does not imply path-connected; the topologist's sine curve is connected yet cannot be traversed by a continuous path.
The set (0, 1) ∪ (2, 3) inside the real line is NOT connected: the two open intervals are disjoint, nonempty, and open (in the subspace), so they split it. By contrast any single interval — open, closed, or half-open — is connected, and the connected subsets of R are exactly the intervals (including unbounded ones and single points).
Two separated intervals form a disconnected set; the connected subsets of the line are precisely the intervals.