Let $X ⊂ \Bbb R^n$ be a non empty subset with $n>0$ and let $x_0 ∈ X$.
Let $Y$ be a non empty topological space and $g : X → Y$ a continuous map.
Suppose $g$ has a continuous extension defined on $\Bbb R^n$.
My question is: Could we affirm that the morphism $g_∗ : π_1 (X, x_0 ) → π_1 (Y, g(x_0 ))$ induced by $g$ is trivial?
I think that $π_1 (\Bbb R^n, x_0 )$ is trivial because $\Bbb R^n$ is simply connected, but we might find a subset $X$ that is not path connected in which case $g_∗(π_1 (X, x_0 ))$ is not trivial?
Thanks for your help.