connectedness
Connectedness asks the simplest possible question about a shape: is it all one piece, or does it fall apart into separate chunks? A single solid blob is connected; two blobs sitting apart are not. The number 8 drawn on paper is connected, but the equals sign = is not — it is two strokes with a gap. This 'all one piece' quality is a topological invariant, so it survives any amount of stretching and bending: you cannot connect or disconnect a shape by deforming it without cutting or gluing.
The precise definition is a clever piece of negative logic. A space X is disconnected if you can split it into two non-empty open sets that do not overlap and together cover X — a clean break into two open halves with nothing in between. X is connected when no such split exists; you simply cannot pry it into two open pieces. A closely related and more visual notion is path-connectedness: X is path-connected if any two of its points can be joined by a continuous path lying inside X — you can walk from anywhere to anywhere without leaving the space. Path-connected always implies connected, and for the everyday shapes (regions, curves, surfaces) the two notions agree.
Connectedness is the first invariant you reach for to tell spaces apart: if one space is in two pieces and another is in one, no homeomorphism can possibly match them, since a homeomorphism preserves the count of pieces. It also underlies the intermediate value theorem — a continuous function on a connected interval cannot jump over a value — which is really the statement that the continuous image of a connected set stays connected. The honest caveat: connected and path-connected are not identical in general (the 'topologist's sine curve' is connected but not path-connected), though the distinction only bites for exotic spaces.
The interval [0, 2] on the line is connected — you cannot split it into two non-empty open pieces. But remove the single point 1 and you get [0, 1) together with (1, 2], two open sets with no overlap covering the rest: now it is disconnected, in two pieces. One missing point changed the answer, which is exactly why connectedness is a delicate, genuinely topological feature.
Deleting one interior point can disconnect a line segment — connectedness is sensitive and topological.
Path-connected implies connected but not conversely; the gap only appears for unusual spaces, so for ordinary regions and surfaces you may treat the two notions as the same.