Graph (topology)

In topology, a subject in mathematics, a graph is a topological space which arises from a usual graph formula_1 by replacing vertices by points and each edge formula_2 by a copy of the unit interval formula_3, where formula_4 is identified with the point associated to formula_5 and formula_6 with the point associated to formula_7. That is, as topological spaces, graphs are exactly the simplicial 1-complexes and also exactly the one-dimensional CW complexes.
Thus, in particular, it bears the quotient topology of the set
under the quotient map used for gluing. Here formula_9 is the 0-skeleton (consisting of one point for each vertex formula_10), formula_11 are the intervals ("closed one-dimensional unit balls") glued to it, one for each edge formula_12, and formula_13 is the disjoint union.
The topology on this space is called the graph topology.
Subgraphs and -trees.
A subgraph of a graph formula_14 is a subspace formula_15 which is also a graph and whose nodes are all contained in the 0-skeleton of formula_14. formula_17 is a subgraph if and only if it consists of vertices and edges from formula_14 and is closed.
A subgraph formula_19 is called a tree iff it is contractible as a topological space.
Applications.
Using the above properties of graphs, one can prove the Nielsen–Schreier theorem.