I have some Algebraic Topology notes that prove the following Lemma:

If $X$ is simply connected and locally path connected, then every covering projection $p:Y \to X$ is trivial (i.e. the whole space $X$ is evenly covered by $p$).

I understand this, and I understand the proof (it constructs a homeomorphism $Y \cong X \times D$ where $D$ is a discrete space). The proof is followed by the comment: "It follows that if a locally path connected space has a simply connected covering space, then the latter is unique up to homeomorphism".

I don't understand why this is true. I can see it's true if we transpose "locally path connected" and "simply connected", since $X$ is lpc iff $Y$ is. Otherwise, I don't really have a clue.

I'd appreciate any enlightenment. Thanks!

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The missing ingredient should be a result on lifting. Say that $p\colon Y\to X$ and $q\colon Z\to X$ are covering projections, with $Y$ and $Z$ simply connected. Since $Z$ is simply connected, $q$ should have a lifting $Z\to Y$, thus making $Z$ a covering space of $Y$. Now apply the lemma. (Commented rather than answering because this may not be the way your textbook argues, and it's been too long since I studied this subject.) – Harald Hanche-Olsen May 21 '12 at 21:00

If $Y,Y'\to X$ are simply connected covering spaces, there is an $X$-morphism $f:Y\to Y'$ (Massey, page 156), which will automatically be a covering space (Massey, p.168). It will be an isomorphism because of the Lemma.