A graph $ \Gamma = \Gamma (V,E) $ consists
- sets $ V , E, $
- A function $E \to E : e \mapsto \bar e$ such that $\bar {\bar e} = e,$ and $\bar e \ne e$ for all $e \in E$
- A function $i : E \to V$ which denotes initial vertex of $i(e)$ of $e.$
Let $E*[0,1]$ denote the disjoint union of the interval $[0,1]$ indexed with $E.$ For simplicity, we denote an element $E*[0,1]$ as $(e,s)$ for $e \in E, s \in [0,1].$ Define an equivalence relation on $V\coprod E*[0,1]$ by taking the smallest equivalence relation such that, for all $e \in E, s \in [0,1]$, $(e,s) \sim (\bar e , 1-s)$ and $(e,0) \sim i(e).$
Define the quotient topology on $\Gamma$ by using above equivalence relation. The above topological spaces can be used to construct spaces whose fundamental group is free groups. For example the figure $8$ can be realised as with $V=\{ * \}$ and $ E= \{ a, \bar a, b,\bar b \} $ with intial vertex of every point in $ E $ is $ * $ and a natural map from $E \to E.$
I am finding tough to find enough exposition on the above topology in literature. In Hatcher, I did not find enough material on this. Where to find material on the above topic?