Math and science::Topology

Connectedness. Equivalent formulations

For a nonempty topological space $X$ , the following are equivalent:

$X$ is connected;
the only subsets that are both open and closed are $\emptyset$ and $X$ ;
every continuous map from $X$ to a discrete space is constant;
every continuous map from $X$ to the two-point discrete space is constant.

The 3rd and 4th statement are similar, yet both useful in their own right: the 3rd statement is general and allows progress to be made in a proof when one knows that $X$ is connected. The 4th is powerful in the reverse direction: simply proving that all continuous maps to the two-point discrete space is constant is enough to show that $X$ is connected.

Context

Source

Leinster, p 69