View source for Connectedness and disconnectedness

Classes of [[topological space]]s 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 space]]s, and the definition of [[totally-disconnected space]] is the disconnectedness corresponding to $\mathcal{P}$ being the class of [[anti-discrete space]]s.  The classes of all topological spaces; of all [[T0 space|$T_0$ space]]s; of all [[T1 space|$T_1$ space]]s and of all one-point spaces are disconnectednesses.  The classes of all one-point spaces; of all [[anti-discrete space]]s; of all [[absolutely connected space]]s; 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}}

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