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## On Fundamental Group - 2

Algebraic Topology
Tsakanikas Nickos
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### On Fundamental Group - 2

Let $X$ be a topological space. Show that if $a,b$ are points in $X$ joined by a curve $\gamma$, then the fundamental groups $\pi_{1}(X,a)$ and $\pi_{1}(X,b)$ are isomorphic. Note that this isomorphism in general depends on the curve $\gamma$. However, show that if $\pi_{1}(X,a)$ is abelian, then $\pi_{1}(X,a)$ and $\pi_{1}(X,b)$ are canonically isomorphic, i.e. the isomorphism does not depend on the curve joining $a$ and $b$.