© 1996, Topology Atlas

The concepts of A-connectedness (also called C(A)-connectedness) and A-disconnectedness were introduced by Gerhard Preuss in his 1967 thesis.
Definitions: Let A be a class of topological spaces. A topological space X is called A-connected if every continuous function from X into a space in A must be constant. A topological space Y is called A-disconnected if every continuous function from an A-connected space to Y must be constant.
Note that if A consists of only the two-point discrete space, then the A-connected spaces coincide with the connected spaces, and the A-disconnected spaces are the totally disconnected spaces, i.e. those spaces whose components are singletons.
The class of A-connected spaces is called a connectedness, and the class of A-disconnected spaces is called a disconnectedness. These classes have also been called left-constant and right-constant.

In addition to Gerhard Preuss, other early investigators of connectedness and disconnectedness include Horst Herrlich, A.V. Arhangel'skii and Richard Wiegandt, and Graciela Salicrup and Roberto Vásquez.

The characterization of connectednesses and disconnectednesses, as well as properties of such classes, soon became an important subject in the study of connectedness and disconnectedness.

The relationship between connectivity and factorizations of continuous functions was pointed out in 1934 by S. Eilenberg and G.T. Whyburn, who independently introduced the (monotone,light) factorization. (Recall that a map is monotone if its fibres are connected, and a map is light if its fibres are totally disconnected.) The (monotone,light) factorization was generalized to other factorization structures, including (A-monotone,A-light) factorization structures, by Peter Collins and Roy Dyckhoff, Graciela Salicrup and Roberto Vásquez, George Strecker, and Walter Tholen.

Graciela Salicrup and Roberto Vásquez extended many of these concepts from the category of topological spaces and continuous functions to Set-based topological categories. They considered factorization structures for single morphisms and for families of morphisms (both sources and sinks), and used them extensively. In Connection and disconnection in LNM 719 (1979) pages 326-344, they extended many of their results to an even larger class of categories which included both topological categories and a large class of Abelian categories, and explored the relationship between connectedness and disconnectedness and torsion theory.

Some Recent Developments

• In 1990, Husek and Pumplun characterized disconnectednesses in a large class of categories in terms of terminal fans, thereby generalizing a result of Arhangel'skii and Wiegandt.
  (see Disconnectedness, Quaestiones Mathematicae, 1990, 13, 449-459.)
• Clementino presented a more general definition of constant morphism in a larger class of categories, and characterized disconnectednesses in terms of multifans.
  (See Constant Morphisms and Constant Subcategories, Appl. Cat. Structures, 1995, 3, no. 2, 119-137.)


• Castellini introduced another notion of constant morphism in order to study connectedness and disconnectedness in categories other than those that are set-based. (See Connectedness, Disconnectedness and Closure Operators, a More General Approach , to appear in the Proceedings of the 1994 L'Aquila Workshop on Categorical Topology.)


• Dikranjan discusses the concepts of connectedness and disconnectedness in relation to topological groups in Compactness and Connectedness in Topological Groups, preprint.

1. Categorical Topology, The Complete Work of Graciela Salicrup, edited by Horst Herrlich and Carlos Prieto; Aportaciones Matemáticas Notas de Investigación No.2; (1988) Sociedad Matemática Méxicana.
   This contains English translations of Salicrup's papers that were written in Spanish, including her thesis.
2. Connectednesses and Disconnectednesses, by H. Lord, Papers on General Topology and Applications: 9th Summer Conference at Slippery Rock University, Annals of the New York Academy of Science, 767 (1995), 115-139.
   This is the first of two survey articles on the topic. It focuses on the early development of the subject.