12.16 in Lee's Introduction To Topological Manifolds Reading through Lee's Introduction To Topological Manifolds. Theorem 12.16 says the following:

Suppose G and H are connected, locally path-connected topological groups, and $\phi:G \to H$ is a surjective continuous homomorphism with discrete kernel. Then if $\phi$ is open or closed, it is a normal covering map.

The proof essentially comes down to this

If $\phi$ is open or closed it is a quotient map and so we have that $H$ is homeomorphic to $G/$ker $\phi$ via a map induced by $\phi$. By an earlier proposition this is then a normal covering map.

Looking through the proof, it appears we only use the fact that $\phi$ is a quotient map not specifically the fact that it is open or closed. Do we actually need the fact that $\phi$ is open or closed, or can we strengthen the theorem by just requiring that $\phi$ is a quotient map?  

You're right, the theorem is still true if you replace the hypothesis that $\phi$ is open or closed by the hypothesis that it's a quotient map. But I wouldn't say that this qualifies as a "strengthening" of the theorem. 
In general, it's harder to prove directly that something is a quotient map than it is to prove that it's open or closed. On the other hand, if you assume that $\phi$ is a quotient map, then it's not hard to prove that it's actually open: Given an open subset $U\subseteq G$, the preimage of $\phi(U)$ is the union of all the open sets $L_k(U)$ as $k$ ranges over $\operatorname{ker}\phi$, hence is open; since $H$ has the quotient topology, it follows that $\phi(U)$ is open in $H$. Thus assuming it's a quotient map doesn't make the theorem apply to any more homomorphisms than it already did.
