Connectedness and disconnectedness

Classes of topological spaces generalising the various notions of connectivity. Let $\mathcal{P}$ be a collection of topological spaces. The connectedness $\mathrm{C}\mathcal{P}$ is the property or class of topological spaces $X$ such that every continuous map from $X$ to any space $P \in \mathcal{P}$ is constant. The disconnectedness $\mathrm{D}\mathcal{P}$ is the dual property or class of topological spaces $X$ such that every continuous map from any space $P \in \mathcal{P}$ to $X$ is constant. Membership of a connectedness or disconnectedness is a topological invariant; membership of a connectedness carries over to topological quotients (is continuously closed) and membership of a disconnectedness carries over to subspaces (is hereditary).

The usual definition of connected space is then the connectedness corresponding to $\mathcal{P}$ being the class of discrete spaces, and the definition of totally-disconnected space is the disconnectedness corresponding to $\mathcal{P}$ being the class of anti-discrete spaces. The classes of all topological spaces; of all $T_0$ spaces; of all $T_1$ spaces and of all one-point spaces are disconnectednesses. The classes of all one-point spaces; of all anti-discrete spaces; of all absolutely connected spaces; and of all topological spaces are connectednesses.

The operators $\mathrm{C}$ and $\mathrm{D}$ establish a Galois correspondence between the systems of classes of continuously closed and hereditary topological spaces.

References

• Arkhangel’skij, A.V.; Wiegandt, Richard "Connectednesses and disconnectednesses in topology" General Topology Appl. 5 (1975) 9-33 DOI 10.1016/0016-660X(75)90009-4 Zbl 0329.54008