We have the theorem which says that the induced homomorphism $p_* : \pi_1(\tilde X,\tilde x_0)\rightarrow \pi_1(X,x_0)$ is injective (hence a monomorphism). Here $\tilde X$ is a covering space of $X$.
I am just trying to digest this fact (the injectivity), since we know many spaces (e.g. wedge of two circles $S^1 \vee S^1$) have covers whose fundamental group is free group on many (or even infinite) generators.
My understanding of Injectivity (as it relates to groups) is that it implies that $\pi_1(X)$ is at least as "big" as $\pi_1(\tilde X)$, so I am confused here.
Confusion: The $\pi_1(X=S^1\vee S^1)$ is free group on two generators. $\pi_1(\tilde X)$ can be free group on many generators. How is the induced map injective then ?
I am not very experienced with group theory, though I have read upon the basics before reading Hatcher/Munkres for Algebraic Topology.