Is the fundamental group of the image a subgroup of the fundamental group of the domain?

Question

I’m trying to learn Algebraic Topology, and there’s a very basic notion that I think must be true, but I can’t seem to prove or disprove it.

Let $f:X\to X$ be a continuous map. Then the fundamental group of the image must be a subgroup of the fundamental group of $X$.

Is this true?

Answer 1

No. Take $f\colon\mathbb{R}^2\longrightarrow\mathbb{R}^2$ defined by $f(x,y)=e^x(\cos y,\sin y)$. Then the fundamental group of the domain is the trivial group, whereas the fundamental group of the image (which is $\mathbb{R}^2\setminus\{(0,0)\}$) is isomorphic to $\mathbb{Z}$.