The Galois correspondence for covering spaces requires that a space $X$ be path-connected, locally path-connected, and semilocally simply connected. These seem like pretty strict requirements, so I was wondering how pathological a space can be while still fulfilling them. More specifically, I was wondering if anyone had an example of a space $X$ fulfilling all three that is not Hausdorff.
Is this even possible? Could we go further and find a space fulfilling these conditions which is not $T_1$?
These conditions are pretty orthogonal to the separation axioms. For instance, any nonempty indiscrete space satisfies these properties. Indeed, any map to an indiscrete space is continuous, so any indiscrete space is contractible (every homotopy you might possibly define is continuous), and locally contractible.
For a less trivial example, consider the space $X=\{a,b,c,d\}$, with the topology generated by the sets $\{a\},\{b,a,c\},\{c\},$ and $\{d,a,c\}$. Each of these basic open sets is contractible: to see that $\{b,a,c\}$ is contractible, for instance, consider the map $H:\{b,a,c\}\times[0,1]\to\{b,a,c\}$ given by $H(x,t)=x$ for $t<1$ and $H(x,1)=b$ (exercise: check that this is continuous). It follows that $X$ is path-connected and locally contractible, so the usual theory of covering spaces applies to $X$. Interestingly, $X$ is not simply connected: in fact, $X$ is weak homotopy equivalent to $S^1$, by a map $S^1\to X$ that sends two points to $b$ and $d$ and the two open arcs between them to $a$ and $c$. So $X$ has a lot of interesting covering spaces. For more discussion see Asphericity of pseudo circle (where I construct the universal cover of $X$) and An interesting topological space with $4$ elements.