Alternate construction of the universal cover of a space Suppose you have a connected, locally path connected Hausdorff space $Y$ that admits a universal covering (i.e. is semilocally simply connected). It occured to me that maybe one can describe the universal covering slightly differently than is usually done (at least from what I have seen). 
Fix a point $x_0\in Y$ and consider the set $$C_{x_0}([0,1]; Y)=\{ \gamma:[0,1]\to Y\textrm{ continuous path }| \gamma(0)=x_0\}$$ which is a subset of $C([0,1];Y)$ equipped with the compact-open topology. Introducing an equivalence $\simeq$ on $C([0,1];Y)$ where $\alpha \simeq \beta$ if $\alpha$ and $\beta$ are endpoint-preserving homotopic, set $$\widetilde Y= C_{x_0}([0,1];Y)/\simeq. $$
This construction certainly agrees as sets with the original (because we simply take the homotopy classes of paths starting at the given fixed point). On the other hand, one may naturally endow $\widetilde Y$ with the quotient topology of  $C_{x_0}([0,1];Y)$ by the equivalence relation $\simeq$.
The question is, does this topology agree with the topology in the original construction? While I'm only really interested in proper geodesic spaces $Y$ I find it interesting that if the above is indeed the 'right' topology then there is a natural constuction that makes sense even if $Y$ does not have a universal cover (what goes wrong in that case is also an interesting question!)
 A: There is a generalised approach to covering spaces, called semicoverings,  using path spaces,  given in this paper of Brazas on semicoverings, which has appeared as Homology, Homotopy and Applications, vol. 14(1), 2012, pp.33–63. Does this suit your needs? 
July 15, 2017 Since the question asks for a "different" description of the construction of the universal cover, I mention that this is given in the bookTopology and Groupoids. This uses covering morphisms of groupoids, and solves the question of when for a topological space $X$ a covering morphism of groupoids $q: G \to \pi_1(X)$, to the fundamental groupoid of $X$,  arises from a covering map $p: Y \to X$. This work first appeared in the 1968 edition of this book, and still seems to me the "right" way to approach the matter, since a map of spaces is modelled by a morphism of groupoids. 
A: Too long for a comment:
I second to read Brazas' paper mentioned in Ronnie Brown's answer, even though strictly speaking that paper does not seem to address the posed question.
I take the usual construction mentioned by the OP to be that of topologizing the (quotient of the) pointed path space by means of the so-called whisker topology, where a whisker neighborhood at (a homotopy class of) a path $\alpha$ is defined to be all (homotopy classes of) paths $\alpha\ast \alpha'$ where the image of $\alpha'$ lies in a neighborhood of $\alpha(1)$.
Notice that Munkres (Topology 2nd edition, Thm 82.1, Step 2 in the proof), just before introducing the whisker topology, mentions the possibility to use the compact-open topology (unfortunately he gives no clue as to why he prefers to proceed with whiskers instead).
I tried to apply the definitions of the two topologies and see whether they agree at least for spaces admitting universal coverings, but I have encountered some difficulties. Namely, I cannot write a basis for the compact-open by using as open sets the "semi-simply connected" open neighborhoods only (that is, the neighborhoods whose inclusion maps induce trivial morphisms on fundamental groups) because in general they are not stable under intersections.
In the arXiv preprint "Topological and uniform structures on universal covering spaces" the authors seem to claim that the whisker and the compact-open topologies agree precisely for what they call "small loop transfer spaces" and, if I am not misunderstanding, this class contains the class of spaces admitting universal coverings. I have not carefully checked the paper though. I hope this helps.
