Projective limit of coverings Let $X$ be a path-connected, semilocally simply connected space.
Let $\tilde X$ be its universal convering and $\Gamma$ the group of deck transformations. Let $(\Gamma_\alpha)_{\alpha\in A}$ be a family of subgroups, closed under finite intersections, such that $\bigcap_{\alpha\in A}\Gamma_\alpha=\{1\}$. Let $X_\alpha=\Gamma_\alpha\backslash \tilde X$ the corresponding covering space and let $\Omega$ be the projective limit over all $X_\alpha$.
If I'm not mistaken, this space is, as a set, the same as $\tilde X$, only equipped with a different topology.
My question is, whether such a space has been investigated anywhere and what is known about it and its topological invariants like homology and homtopy groups.
 A: I'll start by saying that these types of maps $\Omega\to X$ are pretty well-understood even beyond spaces $X$ that admit simply connected covering spaces. Every inverse limit of covering maps (over any space) is a Hurewicz fibration with the unique path-lifting property. This class of fibrations is studied in Spanier's Algebraic Topology book. I am aware of some ongoing research about the geometry of inverse limits of covering maps the focuses on the subtleties around this kind of construction.
Imposing the standard litany of conditions that make a space ``nice" enough to have a universal cover doesn't help. Let's assume that $X$ is path-connected, locally path-connected, and semilocally simply connected so that $X$ has a simply connected covering $p:\widetilde{X}\to X$. Let's say $q:\Omega\to X$ is your inverse limit of covering maps. There is a canonical continuous injection $f:\widetilde{X}\to\Omega=\varprojlim_{\alpha}X_{\alpha}$ such that $q\circ f=p$. You can construct $f$ and verify that it is injective using the lifting properties of the covering maps $X_{\alpha}\to X$ and the universal property of the limit. An inverse limit of path-connected and locally path-connected spaces need not have either property and this is exactly what will happen here. Consider what happens to the fibers. If the spaces $X_{\alpha}$ do not stabilize to $\widetilde{X}$, then $\Omega$ will not be homeomorphic to $\widetilde{X}$ because the fibers of $\Omega\to X$ will be a non-trivial inverse limit of discrete spaces (the fibers of the maps $X_{\alpha}\to X$). However, all fibers of $\widetilde{X}$ are discrete.
However, $f$ is guaranteed to map $\widetilde{X}$ onto a path component of $\Omega$, call it $\widetilde{\Omega}=f(\widetilde{X})$. Moreover, $f:\widetilde{X}\to \widetilde{\Omega}$ is a continuous bijection, which is also a weak homotopy equivalence. This answers your question: the path-components of $\Omega$ are weakly homotopy equivalent to $\widetilde{X}$. The difference is purely topological: $\widetilde{X}$ is the ``locally path coreflection" of $\widetilde{\Omega}$. In particular, $f:\widetilde{X}\to \widetilde{\Omega}$ is a homeomorphism if and only if $\widetilde{\Omega}$ (equivalently $\Omega$) is locally path connected.
Considering that answer, it becomes clear that the utility of $\Omega$ lies outside of spaces with a universal cover. Notice that you don't need for $X$ to admit a universal covering space to construct $\Omega$. You just need enough covering maps over $X$ to distinguish all of the elements of $\pi_1(X)$ from the identity. For instance, if $X$ is the Hawaiian earring or Menger cube, then $\Omega$ as you describe it still exists. Again, inverse limits of path-connected spaces aren't typically path connected. However, the path-components of $\Omega$ are all simply connected. If you pick one, say $\widetilde{\Omega}$, then the restriction $\widetilde{\Omega}\to X$ may not be a true covering map because it may not be locally trivial and $\widetilde{\Omega}$ may not be locally path connected. However,

*

*$\widetilde{\Omega}\to X$ has all of the same lifting properties as a covering map

*$\pi_1(X)$ acts freely and transitively on the fibers of $\widetilde{\Omega}\to X$
In this way, $\widetilde{\Omega}\to X$ serves as a one of a handful of different kinds of "generalized" covering maps.
