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In topology, a branch of mathematics, a graph is a topological space which arises from a usual graph G = {\displaystyle G=} by replacing vertices by points and each edge e = x y ∈ E {\displaystyle e=xy\in E} by a copy of the unit interval I = {\displaystyle I=} , where 0 {\displaystyle 0} is identified with the point associated to x {\displaystyle x} and 1 {\displaystyle 1} with the point associated to y {\displaystyle y}. 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 X 0 {\displaystyle X_{0}} is the 0-skeleton , I e {\displaystyle I_{e}} are the intervals glued to it, one for each edge e ∈ E {\displaystyle e\in E} , and ⊔ {\displaystyle \sqcup } is the disjoint union.
The topology on this space is called the graph topology.