nLab connectology

Idea

A connectology on a set $X$ is a structure which abstracts the information about which subsets of $X$ 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 $X$ consists of a set of subsets of $X$, called connected sets, such that the following axioms hold.

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

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

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

4. If $A,B,\{C_i\}_{i\in I}$ are disjoint connected sets such that $A\cup B\cup \bigcup_i C_i$ is connected, then there is a partition $I=J+K$ such that $A\cup \bigcup_{j\in J} C_j$ and $B\cup \bigcup_{k\in K} C_k$ 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.

Related concepts

• diffeology

References

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