Precise definition of Lifts in Algebraic Topology

Question

I've been reading Algebraic Topology by Allen Hatcher and there are several definitions which bother me due to ambiguity and inconsistency. They are as follows:

1. Is covering map assumed to be continuous and moreover surjective?
2. Is a lift of covering map continuous?

I'm pretty certain about the first one is a "yes", but still has some doubts about the second question.

Note: I'm aware that in Chapter 0 of the book, the author wrote "maps = continuous functions", but I'm also not sure whether it is for Chapter 0 only or for the whole book. I've been reading other reference as well (some lecture notes) to be ensured, but most definitions I found vary.

Answer 1

In short: Yes.

Be careful that sometimes lifts are required to be between pairs (or triplets,...) of spaces, for example $f\colon (X,x_0) \to (Y,y_0)$ continuous maps between pointed spaces.

Furthermore, the concept of lifts in other categories assumes the maps to be morphisms most of the time, i.e. maps that carry over some kind of structural information. Other easy examples are linear maps if we talk about vector spaces, group homomorphisms in group theory, differentiable maps in differential geometry etc.

Answer 2

1. Covering maps $p:E\to B$ are per definition surjective in the book of Munkres. Apparently in the book of Hatcher, they don't have to be surjective as Thorgott explained in the comments of this answer.
2. If $p:E\to B$ is a covering map and $p(e_0)=b_0$, a lift of a continuous map $f:A\to B$ with $f(y_0)=b_0$ is a continuous map $\widetilde{f}$ such that $p\circ\widetilde{f}=f$ and $\widetilde{f}(y_0)=e_0$. This map is unique with the given properties. Path liftings are a special case of this. So yes, they are continuous.