Fundamental group of a tree?

Question

Find the fundamental group of the space $C(T)=\{(x,y) \in T \times T \mid x\neq y\}$.

$C(T)=\{(x,y) \in T \times T \mid x\neq y\}$ where $T$ is a graph

$T$ is the graph made of $3$ edges with a common endpoint below.

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I know two ways to calculate fundamental group:

1. Van Kampen
2. Triagulation $\rightarrow$maximal tree $\rightarrow$ look at generators along 1-simplices

With Van Kampen, I am not sure what $A$ and $B$ will be in this case

And the triangulation method, I am familiar with triangulating, say the Mobius band, and the Klein bottle, but I cannot figure out how to deal with these 3 lines that intersect at a point.... is it already triangulated?

If it is, then we already have a maximal tree, itself. I think this would lead to just one generator element corresponding to fundamental group $\pi = \mathbb{Z}$

Answer 1

You are correct: we have already a maximal tree itself. Now the edges in the complement of this maximal tree generate the fundamental group $\pi$. But... wait! There are no edges in the complement. Hence $\pi= \{1\}$. This amounts to say, that there is no non-trivial loop in $C(T)$.