Connectedness and Disconnectedness
by
Harriet Lord
California State Polytechnic University

In this survey, we look at properties and characterizations of connectednesses and disconnectedness in Top, the category of topological spaces and continuous functions, and determine which of these remain valid in other categories, albeit in a modified form.

These other categories include both topological categories and certain (E, M)-categories.

Some of the properties and characterizations of connectednesses and disconnectednesses in Top that we consider are:

1. Every connectedness is a connection subcategory.
2. Every disconnectedness is a quotient-reflective subcategory.
3. B is a connectedness iff Top has a unique (B-monotone, B-light) factorisation for morphisms whose codomain is T₁.
4. A is a disconnectedness iff A is quotient-reflective and upwards closed.

In addition, we look at the role of closure operators in the study of connectedness and disconnectedness, and also consider various generalizations of the notion of ``constant morphism''.

Definitions: Let K be a ``nice'' category. In the following definitions, we identify full subcategories with their object classes.

Let A and B be subcategories of K. The full subcategories C(A) and D(B) are defined as follows:

C(A) = { X in \CalK | for each A in A every morphism f : X --> A is constant }.

D(B) = { Y in \CalK | for each B in B every morphism f : B --> Y is constant }.

A subcategory B of \CalK is called a connectedness if and only if B = C(D(B)).

A subcategory A of \CalK is called a disconnectedness if and only if A = D(C(A)).

In Top, B is a connection subcategory if it is image invariant, and whenever a class of subspaces of a space X that belong to B has a non-empty intersection its union belongs to B.

A continuous function f is called B-monotone if each of its fibres belongs to B. It is called B-light if each of its fibres belong to D(B).

A subcategory A is called upwards closed if whenever f : X --> Y is an A-monotone surjection with Y in A, then X in A.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # caai-66.