A space falls apart when it can be split into two open pieces, neither empty, sharing no point. It is connected when no such split exists, so it is all one piece.

An island against an archipelago. You can walk anywhere on an island without leaving it; between two islands there is a gap that nothing bridges.

A space falls apart when two open sets, neither empty and sharing no point, account between them for everything in it. The two rings share no point, so an $X$ made only of what they hold is disconnected. Connected means no such split exists anywhere — the space is all one piece.

A separation of $X$ is a pair of open sets, both non-empty, sharing no point, whose union is the whole space. The space is connected when it has no separation. Note that both halves of a separation are closed as well as open, since each is the complement of the other.

• The real line with the point 0 removed is not connected: the negative and the positive numbers separate it.
• A discrete space with two or more points is not connected, since any single point and its complement are both open.
• The indiscrete topology is connected on any set, because the only open sets are $\varnothing$ and $X$ and a separation needs two non-empty ones.
• An interval of the real line is connected. This course states that and does not prove it: the proof needs the least upper bound property, which belongs to the analysis course.

Suppose some set is both open and closed. Its complement is then open too, being the complement of a closed set. A set and its complement share no point and together are everything, so the pair is a separation of the space — unless one of the two is empty. The space is connected, so no separation exists, and therefore one of them must be empty. If the complement is empty the set is everything; if the set is empty it is empty. Those are the only two possibilities, which is what the statement says.

Suppose a subset is open, and closed as well, so that its complement is open too.

A set and its complement together are the whole space, and they share no point.

If both were non-empty the pair would be exactly a separation of the space.

The space is connected, so there is no separation, so one of the two is empty — which leaves only these two possibilities.

And both non-empty, and both open. All four conditions are required.

Two open halves, neither empty, meeting nowhere.

A separation in waiting, unless one of the two is empty.

Both pieces open, so this is a separation.

• A separation is two non-empty open sets sharing no point and covering the space.
• Connected means no separation exists, which has to be argued rather than exhibited.
• Equivalently, nothing is clopen but the empty set and the whole space.
• Intervals are connected; discrete spaces with two or more points are not.