Fork (topology)

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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.