# Are there non-Hausdorff spaces that satisfy the Galois correspondence for covering spaces?

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 $$T_1$$?

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.