Does the torus cover the punctured disc? I am working through Massey's introduction to algebraic topology, < and I have come to the section on covering spaces where it mentions that every induced homomorphism from the fundamental group of a covering space to the fundamental group of the space covered is injective (Chapter 5, section 4, theorem 4.1) My question is then, does the torus cover a punctured disc, and if so, is the induced homomorphism injective? The torus' fundamental group is clearly larger than that of the punctured disc and so I can't reconcile this with theorem 4.1. I can only then assume that the torus doesn't actually cover the punctured disc but I can't see how that is so. Any help with where I have gone wrong in my thinking would be greatly appreciated.

No, it does not. As pointed out in the comments to my question, the covering map between a space and it's covering space has to be a continuous map, which must therefore preserve the compactness of the space. Since a torus is compact and a punctured disc is not, the covering map doesn't preserve compactness and is therefore not a continuous map.

Thank you to the commenters for pointing this out.