Fork (topology)

The notion of a fork appears in the characterization of graphs, including network topology, and topological spaces.

A graph has a fork in any vertex which is connected by three or more edges. Correspondingly, a topological space is said to have a fork if it has a subset which is homeomorphic to the graph topology of a graph with a fork. Stated in terms of topology alone, a topological space "X" has a fork if "X" has a closed subset "T" with connected interior, whose boundary consists of three distinct elements and for which the boundary of the complement of "T" 's interior (relative to "X") consists of these same three elements.

It is perhaps worth noting that certain definitions of a "simple curve" as map "c : I → X" of a real valued interval "I" to a topological space "X" such that "c" is continuous and injective (with the exception, for closed curves, of the two interval endpoints) are "weaker" than the requirement that its range "X" be a connected topological space without forks.