I was thinking about my previous question and thought about going the other way around. Assume we are given a space $Y$ and $Y$ covers $X$, then how much can be said about $X$? The most trivial property would be that if $Y$ is compact, then $X$ would also be compact. In other words, I would like to classify all spaces that a particular space can cover.
Lets pick for example the unit disk $D^1$. Assuming $D^1\to X$ is a cover and $X$ is a CW-complex, then the Euler characteristics gives $\chi(D^1)=1$, so $\chi(X)=1$. The classification theorem would then imply that $X\approx D^1$, since the universal cover of a simply-connected space is just itself.
The second easiest example is probably $\mathbb{C}P^2$. Now the Euler characteristic is $2$, so if it covers a CW-complex $X$, then simple-connectivity would force the cover to have degree $1$ or $2$. In the first case, we would just have the space itself. If the cover has degree $2$ I'm not completely sure what to do, since the only thing we seem to know is that $\pi_1(X)=\mathbb{Z}_2?$
If we drop the assumption that $X$ is a CW-complex, then can anything meaningful be said except that higher homotopy groups are equal?