nLab
connectology

Contents

Idea

A connectology on a set XX is a structure which abstracts the information about which subsets of XX are connected. Every topological space and every graph has an underlying connectology, but not every connectology is of these forms.

Definition

A connectology on a set XX consists of a set of subsets of XX, called connected sets, such that the following axioms hold.

  1. If C𝒫(X)C\subseteq \mathcal{P}(X) is a family of connected sets such that C\bigcap C \neq \emptyset, then C\bigcup C is connected.

  2. Every singleton {x}\{x\} is connected.

  3. If AA and BB are nonempty, and AA, BB, and ABA\cup B are connected, then there exists an xABx\in A\cup B such that A{x}A\cup\{x\} and B{x}B\cup \{x\} are connected.

  4. If A,B,{C i} iIA,B,\{C_i\}_{i\in I} are disjoint connected sets such that AB iC iA\cup B\cup \bigcup_i C_i is connected, then there is a partition I=J+KI=J+K such that A jJC jA\cup \bigcup_{j\in J} C_j and B kKC kB\cup \bigcup_{k\in K} C_k are connected.

A set equipped with a connectology is sometimes called a connective space, although this may be confusing due to other meanings of the word connective.

References

  • Joseph Muscat and David Buhagiar, Connective Spaces, PDF.

Last revised on March 11, 2015 at 08:37:43. See the history of this page for a list of all contributions to it.