Example of bijective continuous map for which the induced map between the fundamental group is not an isomorphism?

Question

Is there any bijective continuous map for which the induced map between the fundamental group is not an isomorphism?

My attempt: If $f: X \rightarrow Y$ is any bijective continuous map, then it is an homeomorphism if $X$ is compact and $Y$ is Hausdorff. And we know that homemorphic spaces have isomorphic fundamental groups. I cannot think of any specific examples.

Answer 1

A simple counterexample is everyone's favorite bijective continuous non-homeomorphism: $$f : [0,1) \to S^1, \quad f(t) = (\cos(2 \pi t), \sin(2 \pi t)) $$

Answer 2

Another counterexample is the identity mapping $I : \Bbb{R} \to \Bbb{R}$ where the topology on the domain is the discrete topology and the topology on the codomain is the usual topology. You can construct a gazillion counterexamples along these lines.