TITLE: Another Question in Hatcher
QUESTION [6 upvotes]: First of all, I apologize for asking yet another question about the hypotheses of a problem in Hatcher, but the statement of one of his problems has stumped me again.
The problem is 1.3.15.  It reads as follows:

Let $p:\widetilde{X}\rightarrow X$ be a simply-connected covering space of $X$ and let $A\subseteq X$ be a path-connected, locally path-connected subspace, with $\widetilde{A}$ a path-component of $p^{-1}(A)$.  Show that $p:\widetilde{A}\rightarrow A$ is the covering space corresponding to the kernel of the map $\pi _1(A)\rightarrow \pi _1(X)$.

My question is:  why do we need to assume that $A$ is path-connected and locally path-connected?
Here is a sketch of the "proof" I have that does not make use of these hypotheses:
If we have a loop in $\widetilde{A}$, this loop is nulhomotopic in $\widetilde{X}$, and hence will be nulhomotpic when mapped down to $X$ via $p$.  On the other hand, if you go the other way and first map the loop in $\widetilde{A}$ down to $A$ via $p$, and then include this loop into $X$, by commutativity of the corresponding diagram I can't draw here, you get the same loop as the first way, which is a nulhomtopic loop.  This shows that $\pi _1[p]\left( \pi _1\left[ \widetilde{A}\right] \right)$ is contained in the kernel of the inclusion.  Convserly, if we have a loop in $A$ that is nulhomotpic in $X$, we can lift this to a loop in $\widetilde{X}$, and hence we can choose such a lift that is contained in $A'$.  Pushing this lift down via $p$, shows that the containment is actually equality.
Any idea where this proof goes wrong?
Thanks in advance!

REPLY [2 votes]: I have just come across this question. It does seem a question where the approach via covering morphisms of groupoids,  as explained  in the 1968, 1988, and 2006 editions of the book now titled and available as "Topology and Groupoids", has clear advantages, since a covering map of spaces induces a covering morphism of groupoids. (This notion occurred under a different name in a 1951 Annals paper of P.A. Smith, and was independently used by Philip Higgins, see his book  Categories and groupoids (downloadable). See also Peter May's book "A concise course in algebraic topology", (1999). It may be conveniently generalised to the notion of fibration of groupoids.)
The pullback of a covering map by a continuous map is also a covering map. The pullback of a covering morphism of groupoids by a morphism of groupoids is also a covering morphism. Such a pullback square can be analysed in detail by a Mayer-Vietoris type sequence, involving object groups and components. (This analysis was not in the 1968 edition of T&G.) 
Massey's book on "Algebraic Topology: an Introduction" also discusses a pullback of covering spaces, if my memory is correct. 
Edit: 15 Feb, 2014 A paper on the Mayer-Vietoris sequence, available here,  is
[40].  (R. Brown,  P.R. HEATH and H. KAMPS), ``Groupoids and the
Mayer-Vietoris      sequence'', J. Pure Appl. Alg. 30 (1983)
109-129.