nLab connectology




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.


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.


  • 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.