Is the induced map on fundamental groups corresponding to inclusion injective?

Question

I know that if $A$ is a subspace of $X$ and $X$ retracts to $A$ then the inclusion induces an injection and the retraction induces a surjection between the fundamental groups. My question is, if no condition of retraction is given, then is it true that the inclusion $A\rightarrow X$ induces an injection? I think it's true and my question may be trivial, but I'd appreciate an answer/explanation.

Answer 1

Let $A = S^1$, $X = \mathbb{R}^2$, and $i : A \to X$ be the inclusion. The map $i_* : \pi_1(A) \to \pi_1(X)$ is not injective. The point is that a non-trivial loop in $A$ may become trivial when considered as a loop in $X$ via the inclusion $i$.