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!)